SAT Triangle Practice: Right Triangles, Similarity & Area

Choose a relationship before choosing a formula

Geometry and Trigonometry is one of the SAT Math content areas listed in College Board's overview. These original examples are for skill practice, not official SAT questions. Solve them yourself, then compare the reasoning.

Example 1: Find the hypotenuse

A right triangle has legs of length 6 and 8. Find its hypotenuse.

Solution: c² = 6² + 8² = 36 + 64 = 100, so c = 10. Use the Pythagorean theorem because the triangle is explicitly right. The hypotenuse is opposite the right angle and must be longer than either leg.

Common error: adding the lengths, or using the theorem on a triangle with no right angle established. The relationship is between squares, and the right-angle condition matters.

Example 2: Scale similar triangles

Two triangles are similar. A side of length 4 in the smaller triangle corresponds to a side of length 12 in the larger. Another smaller side has length 7. Find the corresponding larger side.

Solution: The length scale factor is 12 ÷ 4 = 3. The corresponding side is 7 × 3 = 21. Write which sides correspond before forming the ratio.

Transfer check: If the smaller area is 10 square units, the larger area is 10 × 3² = 90 square units. Areas scale by the square of the length factor, not the length factor itself.

Example 3: Use perpendicular height for area

A triangle has base 12 and perpendicular height 5. Its slanted side has length 13. What is its area?

Solution: A = ½ × base × height = ½ × 12 × 5 = 30 square units. The slanted side is not the height unless it is perpendicular to the chosen base. You do not need all three side lengths here.

Common error: multiplying by 13 simply because it is a labeled length. Identify the perpendicular pair and keep square units in the final answer.

Example 4: Find a missing angle

A triangle has angles of 38° and 67°. What is its third angle?

Solution: The interior angles total 180°, so the missing angle is 180° − 38° − 67° = 75°. Check: 38 + 67 + 75 = 180.

If an exterior angle is shown instead, determine whether the labeled angle lies inside or outside the triangle before subtracting. For example, the exterior angle adjacent to 75° is 105°, because the pair forms a straight angle.

Example 5: A trigonometric ratio

In a right triangle, an acute angle θ has opposite side 9 and hypotenuse 15. Find sin θ.

Solution: sin θ = opposite ÷ hypotenuse = 9 ÷ 15 = 3/5. The opposite side depends on the angle being discussed; the hypotenuse does not. You could find the adjacent side as √(15² − 9²) = 12, making cos θ = 4/5.

What to review next

Record the first step that went wrong. If you chose a correct relationship but made an arithmetic error, add a numerical check. If you matched the wrong sides, redraw the correspondence. If you used a slanted side as height, mark a perpendicular before calculating. Then solve a new triangle problem with different numbers and explain why the chosen relationship applies.