SAT Tip: A Primer on Triangles

SAT Tip: A Primer on Triangles
If you master the concept of "triangle inequality" and the Pythagorean Theorem, SAT problems on triangles will pose few problems (Photograph by Ted VanCleave/Gallery Stock)
Photograph by Ted VanCleave/Gallery Stock

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

Triangles often appear on the SAT, and there are some basic properties you’ll need to know to rock the SAT math section. A triangle is a three-sided figure. The sum of the interior angles is always 180 degrees. To find the area of a triangle, we use the formula A = ½ bh, where b = base and h = height. The base and the height of the triangle must always form a 90-degree angle. Keep in mind that the height can be inside or outside the triangle.

The Triangle Inequality Theorem states that for any side of a triangle, its value must be between the sum and the difference of the other two sides (non-inclusive). For example, if we have a triangle with two sides, 3 and 9, can 6 be the value of the third side? Let’s consider: We know the sum of 3 + 9 is 12, and the difference of 9- 3 is 6, so the third side must be between 6 and 12. Since 6 is not between those two numbers, then 6 cannot be the value of the third side.

The Pythagorean Theorem states that a2 + b2 = c2 where a and b are the two shorter sides and c is always the longest side or the hypotenuse (the side across from the 90-degree angle) of a right triangle. In this triangle, the “?” side is the hypotenuse. Let’s plug the values for the other two sides into the Pythagorean Theorem to solve:

a2 + b2 = c2
82 + 52 = c2
64 + 25 = c2
89 = c2

To remove the exponent (2), we must take the square root (√) of both sides.

√89 = c

Save valuable time on the SAT by memorizing the common Pythagorean triplets. You often encounter right triangles with the ratios of 3:4:5 and 5:12:13. These ratios will also be true for any multiples of 3:4:5 and 5:12:13 such as 6:8:10 or 10:24:26. For example, in this triangle we know the third side must be 5, even without using the Pythagorean Theorem, because we know 5:12:13 is a common triplet.

Be cautious, however, because the 13, or longest side/hypotenuse, must always be across from the 90-degree angle.

There are two special right triangles. The first is a 30-60-90 triangle. Its sides will always be in a ratio of x: x√3 : 2x. The other special triangle is the 45-45-90 triangle. Its sides will always be in a ratio of x: x: x√2. It’s important to remember that for the 30-60-90 triangle, the hypotenuse is the side that has the ratio of 2x. Don’t confuse it with the 45-45-90 ratio and think that the x√3 should be there.

For triangle question practice, take a full-length SAT practice test to sharpen your skills.

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