Decision and conditions
Connect side relationships to areas and angles without trusting the sketch. A right angle or similarity must be given. A 30–60–90 triangle has a hypotenuse twice its shortest side.
Worked model
A right triangle has hypotenuse 15 and one leg 9. What is the other leg? The missing squared leg is 225 − 81 = 144. Take the positive square root, because a side length is positive. The missing leg is 12. Before calculating: Subtract the known leg’s square from the hypotenuse’s square.
Contrast: a different decision
Two similar triangles have corresponding sides 3 and 6. The smaller triangle has area 9. What is the larger area? The length scale from smaller to larger is 2. The area scale is 2² = 4. The larger area is 4(9) = 36. Before calculating: An area scale uses two factors of the length scale.
Transfer with solution
In a 30°–60°–90° triangle, the side opposite 30° has length 5. What is the hypotenuse? The side opposite 30° is the shortest leg. The hypotenuse is twice that leg. Its length is 2(5) = 10. Before calculating: Identify the short leg before using the special right-triangle ratio.
Check the boundary
A linear scale factor is squared for area; an exterior angle is not an interior angle. An exterior angle of a triangle measures 103°. One remote interior angle measures 43°. What is the other remote interior angle, in degrees? The remote angles sum to 103°. Subtract the known remote angle 43°. The other angle is 60°.
A right triangle has hypotenuse 15 and one leg 9. What is the other leg?
- The missing squared leg is 225 − 81 = 144.
- Take the positive square root, because a side length is positive.
- The missing leg is 12.
Compare the other answer choices
Divides the hypotenuse 15 by 3; this does not compute the missing leg.
Subtracts lengths 15−9 instead of their squares.
Adds the two known lengths instead of applying the right-triangle relation.
Reports the missing squared length without taking its square root.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overview