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1.4: Solving Right Triangles

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Learning Objectives

• Solve right triangles.
• Find the area of any triangle using trigonometry.
• Solve real-world problems using right triangles.
• Find the measure of an angle using inverse trig functions.

Inverse Trigonometric Ratios

In mathematics, the word inverse means “undo.” For example, addition and subtraction are inverses of each other because one undoes the other. When we use the inverse trigonometric ratios, we can find acute angle measures as long as we are given two sides.

Inverse Tangent : Labeled \(\tan ^{-1}\), the “-1” means inverse.

\(\tan ^{-1} \left(\dfrac{b}{a}\right)=m\angle B\) and \(\tan ^{-1} \left(\dfrac{a}{b}\right)=m\angle A.\)

Inverse Sine : Labeled \(\sin ^{-1}\).

\(\sin ^{-1} \left(\dfrac{b}{c}\right)=m\angle B\) and \(\sin ^{-1} \left(\dfrac{a}{c}\right)=m\angle A.\)

Inverse Cosine : Labeled \(\cos ^{-1}\).

\(\cos ^{-1} \left(\dfrac{a}{c}\right)=m\angle B\) and \(\cos ^{-1} \left(\dfrac{b}{c}\right)=m\angle A.\)

In most problems, to find the measure of the angles you will need to use your calculator. On most scientific and graphing calculators, the buttons look like \([\sin ^{-1}]\), \([\cos ^{-1}]\), and \([\(\tan ^{-1}]\). You might also have to hit a shift or 2nd button to access these functions.

Now that you know both the trig ratios and the inverse trig ratios you can solve a right triangle. To solve a right triangle, you need to find all sides and angles in it. You will usually use sine, cosine, or tangent; inverse sine, inverse cosine, or inverse tangent; or the Pythagorean Theorem .

What if you were told the tangent of \(\angle Z\) is 0.6494? How could you find the measure of \(\angle Z\)?

Example \(\PageIndex{1}\)

Solve the right triangle.

The two acute angles are congruent, making them both \(45^{\circ}\). This is a 45-45-90 triangle. You can use the trigonometric ratios or the special right triangle ratios.

Trigonometric Ratios

\(\begin{array}{rlrl} \tan 45^{\circ} & =\dfrac{15}{B C} & \sin 45^{\circ} & =\dfrac{15}{A C} \\ B C & =\dfrac{15}{\tan 45^{\circ}}=15 & A C & =\dfrac{15}{\sin 45^{\circ}} \approx 21.21 \end{array}\)

45-45-90 Triangle Ratios

\(BC=AB=15 \text{, } AC=15\sqrt{2} \approx 21.21\)

Example \(\PageIndex{2}\)

Use the sides of the triangle and your calculator to find the value of \(\angle A\). Round your answer to the nearest tenth of a degree.

In reference to \(\angle A\), we are given the opposite leg and the adjacent leg. This means we should use the tangent ratio.

\(\tan A=\dfrac{20}{25}=\dfrac{4}{5}\). So, \(\tan ^{-1} \dfrac{4}{5}=m\angle A\). Now, use your calculator.

If you are using a TI-83 or 84, the keystrokes would be: [2nd][TAN](\(\dfrac{4}{5}\)) [ENTER] and the screen looks like:

Figure \(\PageIndex{4}\)

\(m\angle A \approx 38.7^{\circ}\)

Example \(\PageIndex{3}\)

\angle A is an acute angle in a right triangle. Find \(m\angle A\) to the nearest tenth of a degree for \(\sin A=0.68\), \(\cos A=0.85\), and \(\tan A=0.34\).

\(\begin{aligned} m\angle A&=\sin ^{-1} 0.68\approx 42.8^{\circ} \\ m\angle A&=\cos ^{-1} 0.85\approx 31.8^{\circ} \\ m\angle A&=\tan ^{-1} 0.34\approx 18.8^{\circ} \end{aligned}\)

Example \(\PageIndex{4}\)

Figure \(\PageIndex{5}\)

To solve this right triangle, we need to find \(AB\), \(m\angle C\) and \(m\angle B\). Use only the values you are given.

\(\underline{AB}: \text{ Use the Pythagorean Theorem.}

\(\begin{aligned} 24^2+AB^2&=30^2 \\ 576+AB^2&=900 \\ AB^2&=324 \\ AB&=\sqrt{324}=18 \end{aligned}\)

\(\underline{m\angle B} : \text{ Use the inverse sine ratio.}\)

\(\begin{aligned} \sin B &=\dfrac{24}{30}=\dfrac{4}{5} \\ \sin ^{-1} (45) &\approx 53.1^{\circ} =m\angle B\end{aligned}\)

\(\underline{m\angle C} : \text{ Use the inverse cosine ratio.}\)

\(\cos C=\dfrac{24}{30}=\dfrac{4}{5} \rightarrow \cos ^{-1} (\dfrac{4}{5})\approx 36.9^{\circ} =m\angle C\)

Example \(\PageIndex{5}\)

When would you use sin and when would you use \(\sin ^{-1}\) ?

You would use sin when you are given an angle and you are solving for a missing side. You would use \(\sin ^{-1} \)when you are given sides and you are solving for a missing angle.

Solving the following right triangles. Find all missing sides and angles. Round any decimal answers to the nearest tenth.

Additional Resources

Interactive Element

Video: Introduction to Inverse Trigonometric Functions

Activities: Inverse Trigonometric Ratios Discussion Questions

Study Aids: Trigonometric Ratios Study Guide

Practice: Solve Right Triangles

Angles of Elevation and Depression

You can use right triangles to find distances, if you know an angle of elevation or an angle of depression .

The figure below shows each of these kinds of angles.

The angle of elevation is the angle between the horizontal line of sight and the line of sight up to an object. For example, if you are standing on the ground looking up at the top of a mountain, you could measure the angle of elevation. The angle of depression is the angle between the horizontal line of sight and the line of sight down to an object. For example, if you were standing on top of a hill or a building, looking down at an object, you could measure the angle of depression. You can measure these angles using a clinometer or a theodolite. People tend to use clinometers or theodolites to measure the height of trees and other tall objects. Here we will solve several problems involving these angles and distances.

Finding the angle of elevation

You are standing 20 feet away from a tree, and you measure the angle of elevation to be \(38^{\circ}\). How tall is the tree?

The solution depends on your height, as you measure the angle of elevation from your line of sight. Assume that you are 5 feet tall.

The figure shows us that once we find the value of \(T\), we have to add 5 feet to this value to find the total height of the triangle. To find \(T\), we should use the tangent value:

\(\begin{aligned} tan38^{\circ}&=\dfrac{\text{opposite}}{\text{adjacent}}=\dfrac{T}{20} \\ tan38^{\circ}&=\dfrac{T}{20}\\ T&=20tan38^{\circ}\approx 15.63\\ \text{Height of tree}&\approx 20.63 \text{ ft}\end{aligned}\)

You are standing on top of a building, looking at a park in the distance. The angle of depression is 53^{\circ} . If the building you are standing on is 100 feet tall, how far away is the park? Does your height matter?

Finding the angle of depression

If we ignore the height of the person, we solve the following triangle:

Given the angle of depression is \(53^{\circ}\), \(\angle A\) in the figure above is \(37^{\circ}\). We can use the tangent function to find the distance from the building to the park:

\(\begin{aligned} tan 37^{\circ}=\dfrac{\text{opposite}}{\text{adjacent}}=\dfrac{d}{100}\\ tan37^{\circ} &=\dfrac{d}{100}\\d&=100 tan37^{\circ} \approx 75.36\text{ ft} \end{aligned}\)

If we take into account the height if the person, this will change the value of the adjacent side. For example, if the person is 5 feet tall, we have a different triangle:

\(\begin{aligned} tan37^{\circ}=&\dfrac{\text{opposite}}{\text{adjacent}}=\dfrac{d}{105} \\ tan37^{\circ} &=\dfrac{d}{105} \\ d&=105 tan37^{\circ} \approx 79.12\text{ ft}\end{aligned}\)

If you are only looking to estimate a distance, then you can ignore the height of the person taking the measurements. However, the height of the person will matter more in situations where the distances or lengths involved are smaller. For example, the height of the person will influence the result more in the tree height problem than in the building problem, as the tree is closer in height to the person than the building is.

Real-World Application: The Horizon

You are on a long trip through the desert. In the distance you can see mountains, and a quick measurement tells you that the angle between the mountaintop and the ground is \(13.4^{\circ}\). From your studies, you know that one way to define a mountain is as a pile of land having a height of at least 2,500 meters. If you assume the mountain is the minimum possible height, how far are you away from the center of the mountain?

\(\begin{aligned} tan 13.4^{\circ}&=\dfrac{\text{opposite}}{\text{adjacent}}=\dfrac{2500}{d}\\ tan 13.4^{\circ} &=\dfrac{2500}{d} \\ d&=\dfrac{2500}{tan13.4^{\circ}} \approx 10,494 meters \end{aligned}\)

Example \(\PageIndex{6}\)

You are six feet tall and measure the angle between the horizontal and a bird in the sky to be 40^{\circ} . You can see that the shadow of the bird is directly beneath the bird, and 200 feet away from you on the ground. How high is the bird in the sky?

We can use the tangent function to find out how high the bird is in the sky:

\(\begin{aligned} tan40^{\circ} =\dfrac{height}{200} \\ height&=200 tan40^{\circ} \\ height&=(200)(.839) \\height&=167.8\end{aligned}\)

We then need to add your height to the solution for the triangle. Since you are six feet tall, the total height of the bird in the sky is 173.8 feet.

Example \(\PageIndex{7}\)

While out swimming one day you spot a coin at the bottom of the pool. The pool is ten feet deep, and the angle between the top of the water and the coin is \(15^{\circ}\). How far away is the coin from you along the bottom of the pool?

Since the distance along the bottom of the pool to the coin is the same as the distance along the top of the pool to the coin, we can use the tangent function to solve for the distance to the coin:

\(\begin{aligned} tan15^{\circ}&=\dfrac{\text{opposite}}{\text{adjacent}} \\ tan 15^{\circ} =\dfrac{10}{x} \\ x&=\dfrac{10}{tan15^{\circ}} \\ x&\approx 37.32^{\circ}\end{aligned}\)

Example \(\PageIndex{8}\)

You are hiking and come to a cliff at the edge of a ravine. In the distance you can see your campsite at the base of the cliff, on the other side of the ravine. You know that the distance across the ravine is 500 meters, and the angle between your horizontal line of sight and your campsite is \(25^{\circ}\). How high is the cliff? (Assume you are five feet tall.)

Using the information given, we can construct a solution:

\(\begin{aligned} tan 25^{\circ} =\dfrac{\text{opposite}}{\text{adjacent}}\\ tan25^{\circ}&=\dfrac{height}{500} \\ height&=\dfrac{500}{tan25^{\circ}}\\ height&=(500)(.466) \\ height&=233\text{ meters}\end{aligned}\)

This is the total height from the bottom of the ravine to your horizontal line of sight. Therefore, to get the height of the ravine, you should take away five feet for your height, which gives an answer of 228 meters.

• A 70 foot building casts an 50 foot shadow. What is the angle that the sun hits the building?
• You are standing 10 feet away from a tree, and you measure the angle of elevation to be \(65^{\circ}\). How tall is the tree? Assume you are 5 feet tall up to your eyes.
• Kaitlyn is swimming in the ocean and notices a coral reef below her. The angle of depression is \(35^{\circ}\) and the depth of the ocean, at that point is 350 feet. How far away is she from the reef?
• The angle of depression from the top of a building to the base of a car is \(60^{\circ}\). If the building is 78 ft tall, how far away is the car?

Video: Example: Determine What Trig Function Relates Specific Sides of a Right Triangle

Practice: Angles of Elevation and Depression

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Working with Right Triangles Worksheets

This is the one of the geometric shapes that literally makes the construction of perfectly straight edges possible. Using the Pythagorean theorem, we can find missing sides of these shapes. As we advance into trigonometry, we will learn that you do not much information at all about these shapes to be able to determine all of the angles and lengths of sides. Something that people often do not realize is that if put a diagonal line across a rectangle and cut along that line, you have just created two right-angled triangles. Contractors use the property of geometry all day to create perfectly level and straight structures. This series of lessons and worksheets teach students how to complete figure out all measures on right triangles through a wide variety of techniques.

Aligned Standard: HSG-SRT.C.8

• Find Opposite Step-by-Step Lesson - Given two sides, find the third of the right triangle.
• Guided Lesson - In summer, I always think of problems in my head like number two.
• Guided Lesson Explanation - These problems real help students get the concepts that are mystifying to lower level geometry students.
• Practice Worksheet - This one took me a while. All nicely thought out real world problem set for you to work with.
• Matching Worksheet - In real life if you took the time to think and solve problems like this, you would need to find a hobby.
• Working with Right Triangles Worksheet Five Pack - All these problems have to visualized because they are in sentences.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

There are just so many different ways you can handle these problems.

• Homework 1 - A triangle contains exactly one 90° angle. The other two angles must total exactly 90 degrees. The famous Pythagoras Theorem defines the relationship between the three sides of a right triangle.
• Homework 2 - Jack saw a building that is 75 feet in height. The building casts a 30 foot shadow. What is the angle of elevation from the end of the shadow to the top of the building with respect to the ground?
• Homework 3 - A right triangle's opposite side is 10 and the hypotenuse is 18. Find the value of normal base?

Practice Worksheets

Over the course of the practice I show two different methods for answering the problems.

• Practice 1 - Find the value of normal opposite side?
• Practice 2 - Mark drives 10 km due east of his home. Then he heads 12km north. What is the total distance that he has travelled from his house?
• Practice 3 - Mr. Mike wants to purchase a square shaped table for his office. Table height is 6 feet. Mr. Mike ties a diagonal ribbon on the table. How long must be the ribbon?

Math Skill Quizzes

The first one is just with plain old triangles. The last two are all application word problems.

• Quiz 1 - A right triangle has a opposite side that is 10 units in length and a hypotenuse that is 14 units. Find the value of base?
• Quiz 2 - A road light is 40 feet in height and casts a shadow that is 15 feet long. What is the angle of elevation from the end of the shadow to the top of the road light with respect to the ground? This is assuming that the road light is 100% straight.
• Quiz 3 - Alan has a square shaped television. The height of television is 8 inches. Alan wants to put a diagonal paper on the television. How long must the paper be?

Tips for Working with Right Triangles

A right-angled triangle is one where one of the angles is 90 degrees. As you can see in the diagram to the right, that angle is often denoted by the presence of a square. There are a few unique features of these geometric shapes that you should be aware of. The longest side that is formed as a result of the right angle is opposite that right angle and called the hypotenuse. The altitude in a two-dimensional right triangle is the side that indicates height. The bottom side is referred to as the base. Depending on the orientation the two sides that are not the hypotenuse are interchangeable as result these sides are often referred to as legs. Since a triangle contains 180 degrees of internal angles, the sum of the measures of the legs must be 90 degrees.

The angles are often labeled which allows you to name the triangles. When naming triangles, we can pick any angle name them by labelling the angles in either a clockwise or counterclockwise direction. It does not which you pick as long as the angles fall in same consecutive order that they are found.

Working with right-angled triangles is relatively easier compared to other types of triangles. The reason behind this is simple. You always know the angles, and this can help solve many trigonometric word problems. One of the most common uses of right-angled triangles is the use of the Pythagoras theorem, which is Altitude 2 + Base 2 = Hypotenuse 2 . This allows you to find the measures of missing sides.

When working with right-angled triangles or solving problems that involve them, we can use the properties of sines, cosines, and tangents for determining many different missing sides or angles. Here are some other formulas that can be used with right-angled triangles to identify its unknown parts.

Sines: sin A = a/c, sin B = b/c | Cosines: cos A = b/c, cos B = a/c | Tangents: tan A = a/b, tan B = b/a

Let's just look at some of the cases where we don't know all the sides. Suppose we don't know the sides, but we are familiar with the other two sides. Using the Pythagoras theorem, we can easily identify the remaining two sides. If you know the measure of a side and angle, you can often determine all the others in this geometric environment which makes it a very handy shape to understand.

How Does This Apply to Real Life?

Understanding how to use and manipulate right triangles is one of the most important skills that covered in trigonometry. All of these geometric shapes consist of a single angle that measures ninety degrees. This allows use to use the Pythagorean theorem to our advantage to determine any unknown side or angle within the figure as long as we have a reference side or angle to work off of. In your everyday life you can use this to help achieve very similar things. The minute that you lean a ladder against a straight wall, you have created a right triangle. Want to know the height of the ladder, at what point it touches the wall, or the angle of the ladder, that can all be determined using the same exact math. You can use this math to determine the heights of building or natural structure in much the same way. When we are working with maps to locate areas or plot exact distances between objects, these figures can be pivotal to help you complete this. Anything that involves a coordinate system can be manipulated and better understood with the help of these figures. Mainly because if understand one measure you can learn a great deal about the system it is surrounded in.

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Trigonometry Worksheets

Related Pages Math Worksheets Free Printable Worksheets

There are six sets of Trigonometry worksheets:

• Trig Ratios: Sin, Cos, Tan
• Sin & Cos of Complementary Angles
• Find Missing Sides
• Find Missing Angles
• Area of Triangle using Sine
• Law of Sines and Cosines

Sine, Cosine, & Tangent Worksheets

In these free math worksheets, students learn how to find the trig ratios: sine, cosine, and tangent.

There are five sets of Sine, Cosine, & Tangent worksheets: Trigonometry Worksheet (Learn Adjacent, Opposite, & Hypotenuse) Trigonometric Ratios Worksheets (Sine Ratio, Cosine Ratio, Tangent Ratio)

How to find Sine, Cosine, & Tangent worksheets? In the context of trigonometry, the sides of a right triangle are often described in relation to an angle within the triangle. The common terms used are:

Hypotenuse: The side opposite the right angle. It is the longest side of the right triangle.

Opposite: The side opposite a specified angle. In other words, if you’re looking at one of the non-right angles, the side opposite that angle is called the “opposite” side.

Adjacent: The side adjacent to a specified angle. It is the side that is next to the angle but is not the hypotenuse.

Sine (sin), cosine (cos), and tangent (tan) are three fundamental trigonometric functions that describe the relationships between the sides and angles of a right triangle. These functions are widely used in mathematics and various scientific fields.

Sine (sin): In a right triangle, the sine of an angle (θ) is the ratio of the length of the side opposite the angle to the length of the hypotenuse. sin(θ)= Opposite/Hypotenuse

Cosine (cos): The cosine of an angle (θ) in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. cos(θ)= Adjacent/Hypotenuse

Tangent (tan): The tangent of an angle (θ) in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. tan(θ)= Opposite/Adjacent

These functions are defined not only in the context of right triangles but also more broadly using the unit circle and as periodic functions. They have various applications in physics, engineering, computer science, and other fields where understanding the relationships between angles and sides is crucial.

Click on the following worksheet to get a printable pdf document. Scroll down the page for more Sine, Cosine, & Tangent Worksheets .

More Sine, Cosine, & Tangent Worksheets

Printable (Answers on the second page.) Trigonometry Worksheet #1 (Adjacent, Opposite, & Hypotenuse) Trig Ratios Worksheet #2 (Sin, Cos, Tan Ratios) Trig Ratios Worksheet #3 (Sin, Cos, Tan Ratios) Trig Ratios Worksheet #4 (Sin, Cos, Tan Ratios) Trig Ratios Worksheet #5 (Sin, Cos, Tan Ratios)

Online Trigonometry (sine, cosine, tangent) Trigonometry (sine, cosine, tangent) Trigonometry (using a calculator) Inverse Trigonometry (using a calculator) Trigonometry (find an unknown side) Trigonometry (find an unknown angle) Using Sine Using Cosine Using Tangent Using Sine, Cosine or Tangent Trigonometry Applications Problems Law of Sines or Sine Rule Law of Sines Law of Cosines or Cosine Rule Law of Cosines

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Pythagorean Theorem Worksheet

48 pythagorean theorem worksheet with answers [word + pdf].

The simplicity of the Pythagorean Theorem worksheet is the best thing about it. What is the Pythagorean Theorem? Formulated in the 6th Century BC by Greek Philosopher and mathematician Pythagoras of Samos, Pythagorean Theorem is a mathematic equation used for a variety of purposes. Over the years, many engineers and architects have used Pythagorean Theorem worksheet to complete their projects .

Table of Contents

• 1 Pythagorean Theorem Worksheet
• 2 Knowing Pythagoras of Samos and how he came up with the Pythagorean equation
• 3 Understanding Pythagorean Theorem
• 4 Pythagorean Theorem Word Problems Worksheet
• 5 Using Pythagorean Theorem worksheet
• 6 Conclusion

A simple equation, Pythagorean Theorem states that the square of the hypotenuse (the side opposite to the right angle triangle) is equal to the sum of the other two sides . Following is how the Pythagorean equation is written:

In the aforementioned equation, c is the length of the hypotenuse while the length of the other two sides of the triangle are represented by b and a. Though the knowledge of the Pythagorean Theorem predates the Greek Philosopher, Pythagoras is generally credited for bringing the equation to the fore. This is the reason the Pythagorean equation is named after him. Before we discuss the Pythagorean Theorem and the Pythagorean Theorem worksheet in detail, let’s take a look at who Pythagoras of Samos was and how he came up with the Pythagorean equation.

Knowing Pythagoras of Samos and how he came up with the Pythagorean equation

A 6 th century BC Greek philosopher and mathematician, Pythagoras of Samos is widely credited for bringing the Pythagorean equation to the fore. Though others used the relationship long before his time, Pythagoras is the first one who made the relationship between the lengths of the sides on a right-angled triangle public. This is why he’s regarded as the inventor of the Pythagorean equation.

Apart from being a philosopher and mathematician, Pythagoras founded the Pythagoreanism movement. Born in Croton, Italy, Pythagoras travelled to many different countries including Greece, Egypt, and India. After moving back to Croton in 530 BC, Pythagoras established some kind of school. He returned to Samos in 520 BC. It was in late 6 th Century BC that Pythagoras started to make important contributions to philosophy and math. The Pythagorean equation was one of those contributions.

Though he revealed the Pythagorean equation to the world in the late 6 th Century BC while living in Samos, many historians believe that Pythagoras first thought about the equation during his time in Egypt. In fact, according to many historians, Pythagoras learned geometry, the Phoenicians arithmetic and other branches of mathematics from the Egyptians.

Though he has made many important contributions to philosophy, Pythagoras is widely known as the founder of the Pythagorean Theorem. As previously mentioned, the Pythagorean Theorem is a mathematical equation that states that the square of the hypotenuse (the side opposite to the right angle triangle) is equal to the sum of the other two sides .

Today the aforementioned equation bears Pythagoras’s name but it’s important to know that he wasn’t the first one to use the equation. Before Pythagoras’s time, the Indians and the Babylonians utilized the Pythagorean Theorem or equation. Since they constructed the first proof of the theorem, Pythagoras and his disciples are regarded as the inventors of the equation.

Many historians say that Pythagoras worked in a very secretive manner. This is the reason little evidence is available that the Greek Philosopher/ mathematician himself worked on and proved the Pythagorean Theorem. It is important to note that the first time Pythagoras was given credit for the Theorem was five centuries after his death. This makes Pythagoras’s contribution to the Theorem even more debatable. Nonetheless, since Pythagoras is the only one connected to the Pythagorean Theorem known today, we have to give him due credit. Now that we’ve discussed who Pythagoras of Samos was and how he came up with the Pythagorean equation, it’s time to take a detailed look at the Pythagorean Theorem and the Pythagorean Theorem worksheet.

Understanding Pythagorean Theorem

According to Pythagorean Theorem, the sum of the squares on the right-angled triangle’s two smaller sides is equal to the side opposite to the right angle triangle (the square on hypotenuse). Using a Pythagorean Theorem worksheet is a good way to prove the aforementioned equation. An amazing discovery about triangles made over two thousand years ago, Pythagorean Theorem says that when a triangle has a 90° angle and squares are made on each of the triangle’s three sides, the size of the biggest square is equal to the size of the other two squares put together! A short equation, Pythagorean Theorem can be written in the following manner:

In Pythagorean Theorem, c is the triangle’s longest side while b and a make up the other two sides. The longest side of the triangle in the Pythagorean Theorem is referred to as the ‘hypotenuse’. Many people ask why Pythagorean Theorem is important. The answer to this is simple: you’ll be able to find the length of a right-angled triangle’s third side if you know the length of the other two sides. This equation works like magic and can be used to find any missing value. Following is an example that uses the Pythagorean Theorem to solve a triangle.

In this equation, the longest side of the triangle ‘c’ is missing. By finding out the sum of the squares of the two other sides, we were able to find the missing value. The most famous mathematical contribution of Pythagoras, the Pythagoras Theorem was one of the earliest documented theorems. Though Pythagoras is given most of the credit for the theorem, a major contribution to the theorem was made by his students.

When you look at a Pythagoras Theorem worksheet, you’ll notice that the theorem enables you to find the length of any right angle triangle side provided you know the length of the other two sides. Also, using the theorem, you can check whether a triangle is a right triangle. The Pythagoras Theorem is extremely useful in solving many math problems. Further, you can use it in many real life situations. This is illustrated by a Pythagoras Theorem worksheet.

Pythagorean Theorem Word Problems Worksheet

Using Pythagorean Theorem worksheet

A good way to review the Pythagoras Theorem and expand the mathematical equation is using a Pythagoras Theorem worksheet. By using the worksheet, you’ll be able to get a good understanding of geometry. Additionally, the worksheet will give you an opportunity to review the knowledge related to the different types of triangles. Finally and most importantly, you’ll be able to practice the ancient equation invented by the Greek mathematician and philosopher, Pythagoras. Before you start using the Pythagoras Theorem worksheet, just remember that ‘c’ is the hypotenuse while the shorter sides of the triangle are represented by ‘a’ and ‘b’.

A Pythagoras Theorem worksheet presents students with triangles of various orientations and asks them to identify the longest side of the triangle i.e. the hypotenuse. As you know by now, the formula used in Pythagoras Theorem is a²+b²=c². Regardless of what the worksheet asks the students to identify, the formula or equation of the theorem always remain the same. Though, the students could be presented with different challenges including solving triangles:

• Labeled in different order
• With a different set of letters
• By using vertices to name the sides

The symbols used in the Pythagoras Theorem are something students will find on their calculators. Figuring out how to use these functions is what students need to establish. There is involvement of the Babylonians and the Egyptians in the invention of the Pythagoras Theorem but the earliest known proof of the theorem was produced by the school of Pythagoras.

Many Pythagorean triples were known to the Babylonians while the Egyptians knew and used the (3, 4, 5) triple. The Chinese and Indians also played a role in the invention of the Pythagoras Theorem. The first diagrammatic proof of the theorem was produced by the Chinese while the Indians discovered many triples. In 1995, the theorem became part of the Guinness Book of Records as the most proved theorem of all time.

The triples used in the Pythagoras Theorem include (3, 4, 5), (6, 8, 10), (5, 12, 13), (8,15,17), (7,24,25), (20,21,29), (12,35,37), (9,40,41), (28,45,53), (11,60,61), (16,63,65), (33,56,65) and (48,55,73). The aforementioned triples aren’t multiples of a smaller triple and the name given to them is ‘primitive’ triples. To solve a particular problem, the Pythagoras Theorem can be arranged. For example, if you’re asked to find b which is one of the two smaller sides of the right-angled triangle, you can rearrange the theorem to b²=c²-a². By doing this, you’ll be able to easily find the missing value.

The Pythagoras Theorem has many different proofs. However, when checking your answers, following are the two things that you must always remember:

• The side opposite to the right angle or simply the hypotenuse is always the longest side of the triangle
• Though it is the longest side of the triangle, the size of the hypotenuse can never exceed the sum of the other two squares

To understand this better, take a look at a Pythagoras Theorem worksheet. Today, you can get easy access to Pythagorean Theorem worksheet with answers. Nonetheless, we’re going to try and understand the Pythagoras Theorem as much as we can.

As mentioned earlier, if you know the size of the other two sides, you will be able to find out the length of the third side of the right angle triangle. Also, after being squared, the shorter length is subtracted from the square of the hypotenuse when the hypotenuse is one of the two known lengths. As seen earlier, the lengths of each side of the triangle in the Pythagoras Theorem are whole numbers. Such triangles are known as Pythagorean triangles.

Though there are many different proofs of the Pythagoras Theorem, only three of them can be constructed by students and other people on their own. The first proof starts off as rectangle and is then divided into three triangles that individually contain a right angle. To see first proof, you can use a computer or something as straight forward as an index card cut up into right triangles.

Beginning with a rectangle, the second proof of the Pythagoras Theorem starts off by constructing rectangle CADE with BA=DA. This is followed by the construction of the <BAD’s angle bisector. Once constructed, the bisector is allowed to intersect ED at point F. This makes <BAF and <DAF congruent, BA=DA, and AF=AF. This in turn makes the triangle DAF equal to triangle BAF which means that since ADF is a right angle, ABF will also be a right angle. The third and final proof of the Pythagorean Theorem that we’re going to discuss is the proof that starts off with a right angle. In this proof, triangle ABC is right angle and its right side is angle C.

The three proofs stated above are just few of the many Pythagoras Theorem. You’ll come across these proofs when you take a look at the Pythagorean Theorem worksheet with answers. Learning and understanding the Pythagorean concept is extremely important for students and other people who’ll use this theorem in their practical life.

It is important that you understand the algebraic representation of the Pythagoras Theorem as well as the geometric concepts behind it. You can accomplish this by using proofs, manipulatives, and computer technology. By using these methods to learn Pythagorean Theorem, you’ll be able to see the connections and benefit greatly.

Formulated in the 6th Century BC by Pythagoras of Samos, Pythagoras Theorem is widely used today. If you want to practice Pythagoras Theorem then you can do that easily. Pythagoras Theorem worksheets with answers are easily available and you can use these worksheets to get a good grip of the Theorem.

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High school geometry

Course: high school geometry   >   unit 5.

• Special right triangles proof (part 1)
• Special right triangles proof (part 2)

Special right triangles

• 30-60-90 triangle example problem
• Area of a regular hexagon
• Special right triangles review

• (Choice A)   6 ‍   A 6 ‍  
• (Choice B)   6 3 ‍   B 6 3 ‍  
• (Choice C)   12 ‍   C 12 ‍  
• (Choice D)   18 ‍   D 18 ‍  
• (Choice E)   24 ‍   E 24 ‍  

Right Similar Triangles Worksheet and Answer Key

Students will practice solving for side lengths of right similar triangles .

Example Questions

Visual Aids

Other Details

This is a 4 part worksheet:

• Part I Model Problems
• Part II Practice
• Part III Challenge Problems
• Part IV Answer Key

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