Tip #23 to Get a Top SAT Math Score

In Skills 4 – 7, we looked at finding missing angles in a diagram. Skills 23 – 25 show you how to find a missing side in a diagram. You do not need to memorize these formulas; they are provided in the information box at the beginning of every math section.

I come from a long line of teachers. My grandfather, Herman, was a beloved teacher and principal in Zshmolphf, Austria (mostly true). His father was a scholar (true). His father's father was a scholar. Some say we can trace our lineage back thousands of years to the James Brown of geometry, the godfather of good math, your friend and mine, the mighty Pythagoras (lie), best known for his theorem a² + b² = c². Pythagoras is also definitely a favorite of the folks at the SAT. His theorem is used on EVERY SAT.

Use the Pythagorean theorem, a² + b² = c², to find the length of a side of a right triangle when you are given the lengths of the other two. As I said, this is used on EVERY SAT. In fact, anytime you see a right triangle on the SAT, you will probably need it. Just fill the two sides that you know into the equation, and solve for the third. Here's an example:

If a = 6 and c = 10, what is the value of b ?

The Six Minute Abs of Geometry Length of a Side I

Let's look at this question:

Solution: Great review of SAT Math Mantra #22, "When a picture is described but not shown, draw it." This helps you visualize and organize the info and shows you what to do next. The question does not state whether 10 is another short side like 6 or whether it is the longest side. But we can "Use the Answers" and simply try each answer choice in a² + b² = c², always using the biggest number for c. Choice A is not correct since 3² + 6² ≠ 10², and choice B is not correct since 4² + 6² ≠ 10², and so on. Choice D is correct since 6² + 8² = 10².

Correct answer: D

Example Problems

Easy

The Six Minute Abs of Geometry Length of a Side I

In the right triangle above, if m = 5 and n = 12, find the value of o.
1. 5
2. 12
3. 13
4. 15
5. 21

The Six Minute Abs of Geometry Length of a Side I

If the figure above is a right triangle, what is the value of p ?

Medium

In right triangle ABC (not shown), the measure of AB is 4 and the measure of BC is 10. Which of the following could be the measure of AC ?
4. 84
5. 96

Hard

In an xy coordinate plane (not shown), three points Q, R, and S have coordinates (2, 3), (7, 3), and (2, m), respectively. If the three points determine a right triangle, with a hypotenuse of 13, and m is a positive integer, what is the value of m ?
The lengths of the sides of a right triangle are consecutive integers, and the length of the longest side is x Which of the following could be used to solve for x ?
1. (x – 2)² + (x – 1)² = x²
2. x + x – 3 = x
3. (x – 2)(x – 1) = x²
4. (x – 2) – (x – 1) = x
5. (x – 4)² + (x – 2)² = x²

Answers

C Anytime you see a right triangle on the SAT, try a² + b² = c². Also, for any question with a diagram, mark info from the question into the diagram. Here m and n are the short sides, and o is the longest (also called the hypotenuse), so 5² + 12² = o², so 169 = o² and o= 13.
15 Here 8 and p are the short sides (the longest side is always opposite the right angle). So
B When a picture is described but not shown, draw a diagram. This helps you visualize and organize the info and shows you what to do next. The question does not state whether 10 is another short side like 4 or if it is the longest side. But we can "Use the Answers." Simply try each answer choice along with 10 and 4 in a² + b² = c², always using the biggest number for ^c. To make that easier, convert all the choices to decimals. Choice A is not correct since 4² + ≠ 10². Choice B is correct since 4² + = 10².
15 When a picture is described, but not shown, draw a diagram. This helps you visualize and organize the info and shows you what to do next. Once we see the diagram, piece of cake. We know the hypotenuse is 13, and we see that the length of one side is 5, so 5² + b² = 13². Use algebra to get b = 12. But now don't forget to finish. The length of b is 12, so m = 12 + 3 = 15.
If the longest side is x and the lengths of the sides are consecutive integers, then the shorter sides can be called (x – 1) and (x – 2). Now, we just plug these into a² + b² = c² to get (x – 2)² + (x – 1)² = x².

Go to: Tip #24

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