Tip #25 to Get a Top ACT 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:"

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



2. 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 . Or, if you are given the long side, then the two short sides each measure .

Let's look at this question:

Solution: When you see a right triangle on the ACT, 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, , and 2x for the three sides. Therefore, PR, the shortest side, equals 3. Since the two triangles are congruent, the measure of the shortest side of the other triangle must also be 3.

Correct answer: A

Example Problems

Medium

1. Which of the following sets of 3 numbers could be the side lengths, in meters, of a 30°, 60°, 90° triangle?



2. In the figure below, WXYZ is a square and M, N, O, and P are the midpoints of its sides. If WX = 8 centimeters, what is the perimeter of MNOP, in centimeters?





3. What is the length of the longest leg of a triangle congruent to the triangle shown below?

Hard

4. In an isosceles right triangle, the sum of measures of the two equal sides is 9. What is the measure of the longest side?



5. In a 30, 60, 90 triangle, the sum of measures ofthe shortest and longest sides is 12. What is the measure of the third side?

Answers

1. E This question is rated "medium" only because most kids do not know special right triangles. If you take five minutes right now and memorize them, just the info on the page for Skill 25, then this "medium" becomes easy and you gain points! So once you know about special right triangles, E is obviously the answer. It follows the pattern: x, x, 2x. If, for some crazy reason, you have not memorized the pattern, first of all shame on you, you silly slacker, but second of all, you could draw a diagram for each choice and see which one looks 30-60-90ish.
2. J Great review. You can certainly use special right triangles here, but you could also just "Use the Diagram!" Remember "Use the Diagram"? First you look at it and ask if it seems drawn to scale. Yes, they look like squares, and M, N, O, and P look like midpoints—they are in the middle of each segment. So we can use the picture. If WX is 8, clearly NO is less than 8, perhaps 5 or 6. So the perimeter of MNOP is about 5.5(4) = 22. Convert the answer choices to decimals and choose the best answer. Awesome strategy! The correct answer is J; it's the only one that's even close. Another "medium" made easy! It's so funny that with a in it, an answer choice seems like a foreign language, but translated to decimals, it's so obvious! We can also solve this with special right triangles. Since WXYZ and MNOP are squares, triangle NXO is a 45-45-90 triangle with sides 4, 4, 4 , and the perimeter of MNOP is 4 + 4 + 4 + 4 = 16 .
3. D 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 smaller leg must be 6, and the other leg is 6. A triangle congruent to the one shown would also have a longer leg of 6, since they are congruent.
4. G An isosceles right triangle is a right triangle where two sides are the same length. When two sides are equal (great review of Skill 7), their two opposite angles are also equal. Therefore, we have a 90-45-45 triangle. Since the two equal sides add up to 9, each is 4.5, and the longest side is 4.5.
5. D The sides of a 30-60-90 triangle follow the pattern x, x, 2x. So if the shortest and longest sides add up to 12, then x + 2x = 12. So 3x = 12 and x = 4. Then the other side measures 4. Nice!

Go to: Tip #26