Tip #24 to Get a Top SAT Math Score

When the Pythagorean theorem is not enough to find the length of a missing side, use one or both of the two "special right triangles."

When the three angles of a triangle measure 30°, 60°, 90°, then the sides are x, x, and 2x.

The Six-Minute Abs of Geometry: Length of a Side II

When the three angles measure 45°, 45°, 90° (also called an isosceles right triangle), then the two short sides are equal, let's call them x, and the longest side measures x. Or, if you are given the long side, then the two short sides each measure .

The Six-Minute Abs of Geometry: Length of a Side II

Let's look at this question:

Solution: When you see a right triangle on the SAT, first try the Pythagorean theorem. Given only one side of the triangle, we don't have much info to use in the Pythagorean theorem, so try the special right triangles. Also the 60° angle is a clue to use special right triangles. When you see a 60° angle, try special right triangles. Since there is a 90° and a 60° angle, the third angle must be 30°, so it is a 30, 60, 90 triangle and follows the pattern x, x, and 2x for the three sides. (You do not need to memorize these; they are provided in the information box at the beginning at of every math section.) Therefore, PR = 3 and the longest side of PQR equals 6. Since the two triangles are congruent, the measure of the longest side of MNO must also be 6.

Correct answer: C

Example Problems

Medium

In right triangle MNO (not shown), the measure of MN is 8 and the measure of NO is 4. Which of the following could be the measure of OM ?
1. 4
2. 4
3. 2
1. II only
2. III only
3. I and II
4. II and III
5. I, II, and III

The Six-Minute Abs of Geometry: Length of a Side II

What is the length of the shortest leg of a triangle congruent to the triangle shown above?
1. 3
2. 3
3. 6
4. 6
5. 12

Hard

In an isosceles right triangle, the sum of measures of the two equal sides is 6. What is the measure of the longest side?
1. 3
2. 3 (approximately 4.24)
3. 6
4. 6 (approximately 5.19)
5. 6 (approximately 10.39)

(Question 4 is very similar to a question you've seen before. See if you can solve it this time using special right triangles.)

The Six-Minute Abs of Geometry: Length of a Side II

In the figure above, segments MN and JK are each perpendicular to JM. If x = 30, the measure of LN is 10, and the measure of KL is 6, what is the measure of JM ?
1. 3 (approximately 5.19)
2. 5 (approximately 8.66)
3. 6 (approximately 10.39)
4. 8 (approximately 12.12)
5. 10 (approximately 13.86)

Answers

A 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 8 is another short side like 4 or if it is the longest side. But we can "Use the Answers." Simply try each answer choice in a² + b² = c², always using the biggest number for c. Choice I is not correct since 4² + 4² ≠ 8², choice II is correct since 4² + 4² = 8², and choice III is not correct since 4² + 2² ≠ 8².
C As soon as you see a right triangle with a 60° angle, use the special right triangle 30, 60, 90. The hypotenuse is 12, so the smallest leg must be 6. A triangle congruent to the one shown will also have a shortest leg of 6, since they are congruent.
B An isosceles right triangle is a right triangle where two sides are the same length. When two sides are equal (great review from Skill 6), their two opposite angles are also equal. Therefore, we have a 90, 45, 45 triangle. Since the two equal sides add up to 6, each is 3, and the longest side is
E In Skill 22, we redrew the diagram to scale and estimated the answer. With special right triangles we can get the exact answer. As soon as you see a right triangle with a 30° angle, use the special right triangle 30, 60, 90. We are given the longest side LN = 10, so the side opposite the 30° angle is 5 and LM is 5.

Similarly, since KL = 6, KJ= 3 and JL = 3

Thus, JM = 5 + 3 = 8.

Go to: Tip #25

« prev page
1
2
next page »

View Full Article