SAT Tip: Rocking Right Triangles

Photograph by Don Mason

This tip on improving your SAT score was provided by Vivian Kerr at Veritas Prep.

If you read my earlier post on triangles, you’re now well acquainted with the triangle basics. Let’s look at how we can use our knowledge of special right triangles and the Pythagorean Theorem on SAT geometry questions.

Question 1. Which of the following could represent the side lengths of a right triangle with one angle measuring 30 degrees?

A. 5, 5, 10

B.  5, 5, 5√2

C. 5, 5√2, 5√2

D. 5, 5√2, 5√3

E. 5, 5√3, 10

Since we’re told the triangle is “right,” then one angle is 90 degrees. If one of the other angles is 30 degrees, we’re looking at a 30-60-90 special right triangle. We know that the ratio of a 30-60-90 triangle is x: x√3 : 2x, so the correct answer must have sides in that ratio. Here, x = 5 and the answer is E because it is the only answer choice with the correct ratios of side lengths.

Now let’s look at an example with the 45-45-90 triangle:

Question 2. Which of the following sets of three numbers could be the sides of a right triangle containing a 45 degree angle?

A 1, 1, 1

B 1, 21/2, 21/2

C 2, 2, 2(21/2)

D 1, 21/2, 31/2

E 1, 31/2, 2

This question steps it up a notch by requiring us to know our 45-45-90 side ratio and our exponent rules. Remember that a fractional exponent is just another way of expressing a root. An exponent of ½ is equal to the square root symbol, so 21/2 = √2.

We know the ratio for a 45-45-90 triangle is x: x: x√2, which means two of the sides must be equal. That eliminates D and E. Out of the remaining choices, only (C) correctly expresses the ratio, and it is the correct answer.

Question 3. The restroom in a restaurant is located 5 yards south and 3 yards east from the hostess stand, and Joe’s table is 2 yards north and 6 yards west of the hostess stand. Approximately how far is Joe from the restroom?

Start by drawing the restaurant, as described. We want to find the distance between Joe’s table and the restroom. We can draw a triangle so that this distance is the hypotenuse of a triangle.

The horizontal distance from Joe to the restroom is 6 + 3, so one leg of the triangle is 9. The vertical distance from Joe to the restroom is 2 + 5, so the other leg of the triangle is 7. Now that we know two legs of a right triangle, we can solve using the Pythagorean Theorem:

a2 + b2 = c2

72 + 92 = c2

49 + 81 = c2

130 = c2

11.4 ≈ c

The answer is 11 yards, approximately. This question didn’t use our Pythagorean triplets—common ratios for right triangles that will allow you to answer many questions like this one without using the Pythagorean Theorem. Still, remember the two most common ratios—3:4:5 and 5:12:13—and look out for them hiding in geometry questions on test day.

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