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## SAT

### Course: SAT>Unit 6

Lesson 6: Additional Topics in Math: lessons by skill

# Right triangle word problems | Lesson

## What are right triangle word problems, and how frequently do they appear on the test?

Right triangle word problems on the SAT ask us to apply the properties of right triangles to calculate side lengths and angle measures.
In this lesson, we'll learn to:
1. Use the Pythagorean theorem and recognize Pythagorean triples
2. Use trigonometric ratios to calculate side lengths
3. Recognize special right triangles and use them to find side lengths and angle measures
On your official SAT, you'll likely see 1 question that is a right triangle word problem.
This lesson builds upon the Right triangle trigonometry skill.
You can learn anything. Let's do this!

## How do I calculate side lengths using the Pythagorean theorem?

### Intro to the Pythagorean theorem

Khan Academy video wrapper
Intro to the Pythagorean theoremSee video transcript

### The Pythagorean theorem

In a right triangle, the square of the hypotenuse length is equal to the sum of the squares of the leg lengths.
At the beginning of each SAT math section, you'll find this diagram provided as reference:
c, squared, equals, a, squared, plus, b, squared

#### Calculating missing side lengths in right triangles

With the Pythagorean theorem, we can calculate any side length in a right triangle when given the other two.

Let's look at some examples!
What is the length of start overline, B, C, end overline in the figure above?

What is the length of start overline, E, F, end overline in the figure above?

#### Recognizing Pythagorean triples

Pythagorean triples are integers a, b, and c that satisfy the Pythagorean theorem. For example, the side lengths of the right triangle shown below form a Pythagorean triple:
Each side of the triangle has an integer length, and 5, squared, equals, 3, squared, plus, 4, squared. 3-4-5 is the most commonly used Pythagorean triple on the SAT. All triangles similar to it also have side lengths that are multiples of the 3-4-5 Pythagorean triple, like 6-8-10, 9-12-15 or 30-40-50.
Being able to recognize Pythagorean triples can save you valuable time on test day. For example, if you see a right triangle with a hypotenuse length of 15 and a leg length of 12, recognizing it's a 9-12-15 triangle will give you the missing side length, 9, without having to calculate it using the Pythagorean theorem.
Less frequently used Pythagorean triples include 5-12-13 and 7-24-25.

### Try it!

try: use pythagorean triples and similarity to find side lengths
In the figure above, start overline, B, E, end overline is parallel to start overline, C, D, end overline.
What is the length of start overline, B, E, end overline ?
If C, D, equals, 6, what is the length of start overline, A, C, end overline ?

## How do I use trigonometric ratios and the properties of special right triangles to solve for unknown values?

### Recognizing side length ratios

#### Using trigonometric ratios to find side lengths

Sine, cosine, and tangent represent ratios of right triangle side lengths. This means if we have the value of the sine, cosine, or tangent of an angle and one side length, we can find the other side lengths.

Let's look at an example!
In the figure above, tangent, left parenthesis, C, right parenthesis, equals, start fraction, 4, divided by, 7, end fraction. What is the length of start overline, A, C, end overline ?

#### Using special right triangles to determine side lengths and angle measures

Special right triangles are right triangles with specific angle measure and side length relationships. At the beginning of each SAT math section, the following two special right triangles are provided as reference:
This means when we see a special right triangle with unknown side lengths, we know how the side lengths are related to each other. For example, if we have a 30, degrees-60, degrees-90, degrees triangle and the length of the shorter leg is 3, we know that the length of the hypotenuse is 2, left parenthesis, 3, right parenthesis, equals, 6 and the length of the longer leg is 3, square root of, 3, end square root.
We can also identify the angle measures of special right triangles when we spot specific side length relationships. For example, if we're given a right triangle with identical leg lengths, we know it's a 45, degrees-45, degrees-90, degrees special right triangle.

### Try it!

try: recognize trigonometric ratios and special right triangles
Right triangle A, B, C is shown in the figure above. The value of sine, left parenthesis, A, right parenthesis is start fraction, 1, divided by, 2, end fraction.
What is the length of start overline, A, C, end overline ?
What is the measure of angle C ?
degrees

## Your turn!

Practice: find segment length
In the figure above, start overline, B, D, end overline is parallel to start overline, A, E, end overline. What is the length of start overline, D, E, end overline ?

Practice: find side length
In the figure above, sine, left parenthesis, C, right parenthesis, equals, start fraction, 3, divided by, 5, end fraction. What is the length of start overline, B, C, end overline ?

Practice: find angle measure
In quadrilateral A, B, C, D above, start overline, A, D, end overline is parallel to start overline, B, C, end overline and C, D, equals, square root of, 2, end square root, A, B. What is the measure of angle C ?
Choose 1 answer:

## Want to join the conversation?

• In the last example, What do you mean by 'SAT provides us'? Does the question provide it is a 45-45-90 triangle. If it does not provide, it could also be 30-60-90 triangle. How do we know?
(3 votes)
• The explanation is talking about the formula sheet that you get to reference during the math sections. It contains things like area and volume formulas, as well as the side-length ratios for 45-45-90 and 30-60-90 right triangles. So, if you didn't feel like memorizing the ratios, or forgot in the heat of the moment, you can always check the formula sheet at the front of the section.
From the information that the question gives you, you can conclude that the triangle is 45-45-90. The hypotenuse of the right triangle is sqrt(2) * AB, which is the same length as one of the legs of the triangle. Combine this with the information that the triangle is a right triangle, and you have enough information to be able to state that the triangle is 45-45-90, and thus know all of its angle measures.
(10 votes)
• For the SAT, would the right triangle word problems be in the calculator or non-calculator section?
(4 votes)
• I need help on the topics of
Recognizing Pythagorean Triples
Recognizing side length ratios
Using trigonometric ratios to find side lengths
Using special right triangles to determine side lengths and angle measures
(1 vote)