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Right Triangles

Rules, Formula and more

Pythagorean Theorem

The sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse .

Usually, this theorem is expressed as $$ A^2 + B^2 = C^2 $$ .

Right Triangle Properties

Right triangle picture

A right triangle has one $$ 90^{\circ} $$ angle ($$ \angle $$ B in the picture on the left) and a variety of often-studied formulas such as:

  • The Pythagorean Theorem
  • Trigonometry Ratios (SOHCAHTOA)
  • Pythagorean Theorem vs Sohcahtoa (which to use)

SOHCAHTOA only applies to right triangles ( more here ) .

sohcahtoa

A Right Triangle's Hypotenuse

The hypotenuse is the largest side in a right triangle and is always opposite the right angle.

Hypotenuse

In the triangle above, the hypotenuse is the side AB which is opposite the right angle, $$ \angle C $$.

Online tool calculates the hypotenuse (or a leg) using the Pythagorean theorem.

Practice Problems

Below are several practice problems involving the Pythagorean theorem, you can also get more detailed lesson on how to use the Pythagorean theorem here .

Find the length of side t in the triangle on the left.

5, 12, 13 right triangle

Substitute the two known sides into the Pythagorean theorem's formula : A² + B² = C²

What is the value of x in the picture on the left?

pythagorean theorem

Set up the Pythagorean Theorem : 14 2 + 48 2 = x 2 2,500 = X 2

$$ x = \sqrt{2500} = 50 $$

Diagram, Pythagorean Theorem

$$ x^2 = 21^2 + 72^2 \\ x^2= 5625 \\ x = \sqrt{5625} \\ x =75 $$

Find the length of side X in the triangle on on the left?

3, 4, 5 right triangle

Substitue the two known sides into the pythagorean theorem's formula : $$ A^2 + B^2 = C^2 \\ 8^2 + 6^2 = x^2 \\ x = \sqrt{100}=10 $$

What is x in the triangle on the left?

pythagorean image

x 2 + 4 2 = 5 2 x 2 + 16 = 25 x 2 = 25 - 16 = 9 x = 3

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Surface area of a Cylinder

Triangle Solving Practice

Practice solving triangles .

You only need to know:

  • Angles Add to 180°
  • The Law of Sines
  • The Law of Cosines

Try to solve each triangle yourself first, using pen and paper.

Then use the buttons to solve it step-by-step (more Instructions below).

Instructions

  • Look at the triangle and decide whether you need to find another angle using 180°, or use the sine rule, or the cosine rule.  Click your choice .
  • The formula you chose appears, now click on the variable you want to find.
  • The calculation is done for you.
  • Click again for other rules until you have solved the triangle.

Note: answers are rounded to 1 decimal place.

What Does "AAS", "ASA" etc Mean?

It means which sides or angles we already know:

Free Mathematics Tutorials

Free Mathematics Tutorials

Right triangle questions.

Multiple choice questions right triangle problems related to trigonometry with answers at the bottom of the page.

Questions with their Answers

Pin it!

Question 10

Question 11, question 12, more references and links, popular pages.

  • Trigonometry Problems and Questions with Solutions - Grade 10
  • Questions on Angles in Standard Position
  • Free Trigonometry Tutorials and Problems
  • Solve Trigonometry Problems
  • Free Trigonometry Worksheets to Download

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10 2 solving right triangles practice problems

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Mathematics LibreTexts

1.4: Solving Right Triangles

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  • Page ID 61221

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.

f-d_2a337f0fc303f5ead45b9c00ac0b99fc56d2875b59312a4bb58a17d7+IMAGE_TINY+IMAGE_TINY.png

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.

Screen Shot 2021-04-16 at 13.28.47.png

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.

Screen Shot 2021-04-16 at 15.09.57.png

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:

Screen Shot 2021-04-16 at 15.10.24.png

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 \(\PageIndex{4}\)

Screen Shot 2021-04-16 at 15.10.50.png

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.

Screen Shot 2021-04-16 at 15.16.26.png

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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High school geometry

Course: high school geometry   >   unit 5.

  • Triangle similarity & the trigonometric ratios

Trigonometric ratios in right triangles

  • (Choice A)   3 5 ‍   A 3 5 ‍  
  • (Choice B)   4 5 ‍   B 4 5 ‍  
  • (Choice C)   3 4 ‍   C 3 4 ‍  
  • (Choice D)   4 3 ‍   D 4 3 ‍  

IMAGES

  1. Understanding Trigonometry Grade 10

    10 2 solving right triangles practice problems

  2. How To Solve A Right Triangle For Abc / Solving Triangles

    10 2 solving right triangles practice problems

  3. 10++ Solving Right Triangles Worksheet Answers

    10 2 solving right triangles practice problems

  4. Special right triangles quiz

    10 2 solving right triangles practice problems

  5. 12.2

    10 2 solving right triangles practice problems

  6. Solving Right Triangle Applied Problems 5 10

    10 2 solving right triangles practice problems

VIDEO

  1. Trigonometry Solving Right Triangles Day 2 2024

  2. Solving Right Triangles (part I)

  3. Solving Right Triangles

  4. 130.032.1 Solving Right Triangles 3

  5. 5 .2 Part 3( Solving triangles)

  6. 2 SPECIAL Triangles that EVERY Math student must absolutely know!

COMMENTS

  1. Solve for a side in right triangles (practice)

    High school geometry > Right triangles & trigonometry > Solving for a side in a right triangle using the trigonometric ratios Solve for a side in right triangles Google Classroom You might need: Calculator B C = Round your answer to the nearest hundredth. 35 ∘ ? 6 C B A Show Calculator Stuck? Review related articles/videos or use a hint.

  2. PDF Right Triangle Trig Missing Sides and Angles

    10.8 A B C 57° 12.9 15) 10.3 x A C B 37° 6.2 16) 3 x A C B 47° 2 Solve each triangle. Round answers to the nearest tenth. 17) 22.6 mi B A C 62° 28° 12 mi 25.6 mi 18) 9 in B C A 51° 39° 7.3 in 11.6 in 19) A 4.5 mi B C 42° 48° 5 mi 6.7 mi 20) A 5 m B C 37° 53° 4 m 3 m 21) 29.3 mi B A C 62° 28° 15.6 mi 33.2 mi 22) 14 mi A B C 24° 66 ...

  3. Right Triangles, Hypotenuse, Pythagorean Theorem Examples and Practice

    Diagram 1 Diagram 2 Right Triangle Properties A right triangle has one 90∘ 90 ∘ angle ( ∠ ∠ B in the picture on the left) and a variety of often-studied formulas such as: The Pythagorean Theorem Trigonometry Ratios (SOHCAHTOA) Pythagorean Theorem vs Sohcahtoa (which to use) SOHCAHTOA only applies to right triangles ( more here) . Picture 2

  4. Solving Right Triangles: Problems

    Study Guide Topics Solving Right Triangles Right Triangle Review Techniques for Solving Problems Applications Problems Terms and Formulae Problems Previous Next Problem : Solve the following right triangle, in which C = 90o: a = 6, B = 40o . A = 90o - B = 50o. b = a tan (B) 5.0. c = 7.8 .

  5. Triangle Solving Practice

    Instructions Look at the triangle and decide whether you need to find another angle using 180°, or use the sine rule, or the cosine rule. Click your choice. The formula you chose appears, now click on the variable you want to find. The calculation is done for you. Click again for other rules until you have solved the triangle.

  6. Solving a Right Triangle Practice

    Solving a Right Triangle Geometry Skills Practice 1. Which corresponds to the approximate measure of side SM for the triangle drawn below? Simplify your answer to one decimal place only. 2....

  7. IXL

    IXL plans. Virginia state standards. Textbooks. Test prep. Awards. Improve your math knowledge with free questions in "Solve a right triangle" and thousands of other math skills.

  8. IXL

    out of 100. IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. It tracks your skill level as you tackle progressively more difficult questions. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)! Learn more.

  9. 5.2: Solution of Right Triangles

    27.47 is obtained by long division: If you are using a pocket calculator, type. Answer: x = 27.5. There is an easier method to solve Example 5.2.4. ∠B = 90 ∘ − 20 ∘ = 70 ∘ The leg opposite ∠B is x and the leg adjacent to ∠B is 10 . tan70 ∘ = opp adj 2.7475 = x 10 (2.7475)(10) = x 27.475 = x 27.5 = x.

  10. 2.2: Solving Right Triangles.

    Skills. Practice each skill in the Homework Problems listed. 1 Solve a right triangle #1-16, 63-74. 2 Use inverse trig ratio notation #17-34. 3 Use trig ratios to find an angle #17-22, 35-38. 4 Solve problems involving right triangles #35-48. 5 Know the trig ratios for the special angles #49-62, 75-78.

  11. Solve triangles using the law of sines (practice)

    Precalculus > Trigonometry > Law of sines Solve triangles using the law of sines Google Classroom You might need: Calculator The following figure shows A B C with side lengths to the nearest tenth. 22 ° 25 10 B A C Find m ∠ C . Note that m ∠ C is obtuse. Round to the nearest degree. m ∠ C = ° Show Calculator Stuck?

  12. Solve Right Triangle Problems

    Right triangles problems are solved and detailed explanations are included. Example - Problem 1: Find sin (x) and cos (x) in the right triangle shown below. Solution to Problem 1: First use the Pythagorean theorem to find the hypotenuse h of the right triangle. h = √ (6 2 + 8 2 ) = √ (36 + 64) = 10

  13. PDF 10.2 Right Triangle Similarity Word Problems

    _____ can solve similar right triangle word problems! 10.2 - Right Triangle Similarity Word Problems Find the value of x and y: In the diagram below of right triangle AED, ∥ . \ Which statement is always true? Let's review… triangle similarity Similar triangles have _____ sides and _____ angles.

  14. Right Triangle Questions

    Find the length of AC in the right triangle below. a) 9 b) 9 √2 c) 18 √2 d) 18 Question 7 Find the length of the hypotenuse in the right triangle below where x is a real number. a) 5 b) 10 c) 25 d) √ 5 Question 8 Find the area of a square whose diagonal is 40 meters. a) 80 m 2 b) 800 m 2 c) 1600 m 2 d) 40 m 2. Question 9

  15. 1.4: Solving Right Triangles

    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.

  16. Right triangle trigonometry word problems (practice)

    High school geometry > Right triangles & trigonometry Modeling with right triangles Right triangle trigonometry word problems Google Classroom You might need: Calculator Bugs Bunny was 33 meters below ground, digging his way toward Pismo Beach, when he realized he wanted to be above ground.

  17. Special Right Triangle Practice

    nearest square inch . 6. The altitude of an equilateral triangle is 9 inches long. Find the perimeter of the triangle. 7. In the diagram below, points A, B, C and D are collinear. Find x and y. 8. Martin walks his dog on level ground in a straight line with the dog's favorite tree.

  18. Pythagorean Theorem Practice Problems With Answers

    Answer Problem 2: Find the value of [latex]x [/latex] in the right triangle. Answer Problem 3: Find the value of [latex]x [/latex] in the right triangle. Answer Problem 4: The legs of a right triangle are [latex]5 [/latex] and [latex]12 [/latex]. What is the length of the hypotenuse? Answer

  19. Trigonometric ratios in right triangles (practice)

    Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. ... A right triangle A B C. Angle A C B is a right angle. Angle A B C is beta. Side A C is three units. ... Report a problem. Stuck? Review related articles/videos or use a hint.

  20. Right Triangle Trigonometry

    Solve each problem. See Examples 3 and 4. The figure to the right indicates that the equation of a line passing through the point (a, 0) and making an angle θ with the x-axis is y = (tan θ) (x - a). Find an eq uation of the line passing through the point (5, 0) that makes an angle of 15° with the x-axis. 27.