Question 560·Easy·Math

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In a right triangle, one leg measures $6$ centimeters and the hypotenuse measures $10$ centimeters. What is the length, in centimeters, of the triangle's other leg?

The cue for the Pythagorean theorem is a right triangle with a missing side: the legs’ squares add to the hypotenuse’s square. Put the hypotenuse, the side across from the right angle, alone on one side of the equation. For a missing leg, subtract the known leg’s square, then use a Desmos regression to find the positive length. Subtracting the side lengths themselves misses the squared relationship.

Hints

Hint 1
The hypotenuse is the side across from the right angle. Its square equals the sum of the two legs’ squares. Which measurement belongs alone on one side of that equation?
Hint 2
Because you need a leg, subtract the known leg’s square from the hypotenuse’s square. That gives the missing leg’s square, not its length. How can you find the positive length?

Step-by-step

Approach 1: Use the Pythagorean theorem
Step 1
Write the equation for the missing leg
The problem calls $10$ the hypotenuse, the side across from the right angle, and $6$ a leg. Let $x$ be the other leg. The Pythagorean theorem says the legs’ squares add to the hypotenuse’s square, so $x^2+6^2=10^2$ .
Step 2
Isolate the missing leg’s square
Subtract $6^2$ from both sides to leave the missing leg’s square: $x^2=10^2-6^2$ . For a missing leg, subtract squares, not side lengths.
Step 3
Find the positive length
Type $x_1^2\sim10^2-6^2\{x_1>0\}$ in Desmos. Use $x_1$ instead of $x$ so Desmos solves rather than graphs; $\sim$ asks its regression to make the sides equal. The positive-only restriction matters because a length can’t be negative. Under PARAMETERS, Desmos shows $x_1=8$ , so the other leg is $8$ centimeters. Choice C.

Approach 2: Recognize a scaled right triangle
Step 1
Match the known sides to a pattern
A Pythagorean triple is a set of whole-number sides of a right triangle. The $3$ - $4$ - $5$ triple works because $3^2+4^2=5^2$ . Here $6=2\cdot3$ and $10=2\cdot5$ , so the known sides are twice the matching triple sides.
Step 2
Scale the remaining leg
Scaling a triangle multiplies every side by the same number, so the remaining leg is $2\cdot4=8$ . The other leg measures $8$ centimeters. Choice C.

Lessons that teach this

SAT Geometry and TrigonometryBeginnerCoreApply the Pythagorean theorem

Question 560·Easy·Math

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