# Right Triangles on the PSAT/NMSQT

You’ll find lots of triangles on the PSAT/NMSQT, especially right triangles. The Greeks weren’t the only mathematicians in the ancient world, but they managed to place their “brand” on *geometry,*** **a word which, by the way, comes from the Greek words for “earth measure.” Specifically, a mathematician named Pythagoras wrote the Pythagorean Theorem:

a^{2}+b^{2}=c^{2}

You can use this formula to find the sides of any right triangle, in which *a *and *b *are defined as the two legs of the triangle and *c *is the ** hypotenuse,** a fancy word for the side opposite the 90̊ angle.

**This formula — the Pythagorean Theorem — appears in the information box on the exam.**

*Note:*A few common right-triangle ratios are frequent fliers on the PSAT/NMSQT, so it’s worth memorizing them:

**The 3:4:5 triangle:**The sides can be any multiple of these numbers (for example, 15:20:25, with each side multiplied by 5, or 21:28:35, with each side multiplied by 7).**The 5:12:13 triangle:**Strange numbers, huh? But this ratio behaves like any other, so you can multiply each side by 2 and get a 10:24:26 triangle, or multiply by 5 and get a 25:60:65 triangle.**The****triangle:**The*s*stands for a side, and because you have two sides that are equal (both are*s*), this is anand the interior angles are 45°, 45°, and 90°.*isosceles right triangle,*This formula appears in the information box on the exam.*Note:*You can use the information in the preceding bullet to calculate the

(a line connecting opposite corners) of a square. If the sides of a square are 65 meters long, the diagonal is*diagonal*You can easily see why this formula works: A square is just two isosceles right triangles glued together, because each side of a square is the same length.

**The****triangle:**This one has 30°-60°-90° angles, and for some reason, the exam-writers love it. The hypotenuse (the long side) is double the length of the side opposite the 30º angle.This formula appears in the information box on the exam.*Note:*If you cut an

(one with equal sides) in half, you get two 30°-60°-90° triangles. So if you see a question on the exam about an equilateral triangle, drag out this formula and you’ll find the answer in a flash.*equilateral triangle*

Stretch those triangular muscles! Try these problems:

An equilateral triangle has a side with a length of

*x.*What is the area of the triangle, in terms of*x?*What is the area of the following figure?

(A) 6

(B) 15

(C) 32

(D) 36

(E) 42

In the following square, the product of diagonals

*AC*and*BD*is 18. What is the perimeter of triangle*ABC?*(C) 18

(D) 24

(E) 36

Now check your answers.

B.

Always draw a picture if you’re having any trouble visualizing a problem:

Remember that you can transform all equilateral triangles into a pair of 30º-60º-90º triangles by cutting them in half. That lets you see that the base of one of the smaller triangles is

*x*/2, and the height ismaking the area of the whole triangle equal

**D.**36How well do you know your Pythagorean triples? Either way, you can use the Pythagorean Theorem to help you solve this problem, or any problem with right triangles. First, look at the small triangle. Its area is

Next, you can quickly solve for the hypotenuse using

*a*^{2}+*b*^{2}=*c*^{2},*b*= 12 and determine that it’s 5 units long.Pythagorean theorem to the rescue again: 5

^{2}+*b*^{2}= 13^{2},*b*= 12. That means that the area of the larger triangle isAdd those two areas together: 6 + 30 = 36, and you can see that Choice (D) is correct.

**A.**Diagonals

*AC*and*BD*must be the same length, so they’re eachin length. Now you can ignore the square and just pay attention to triangle

*ABC,*which is a 45̊-45̊-90̊ triangle, with a hypotenuse ofUsing your knowledge of special triangles (or the formula box), you know that the legs of the triangle must each be 3 units long. Therefore, the perimeter of the triangle is