Question

A ladder is leaning against a wall to form a triangle with the ground and the wall. The length of the ladder is 12 feet. The distance from the wall to the base of the ladder is 6 StartRoot 2 EndRoot Feet. The wall and the floor form a right angle.
A 12-foot ladder is leaning against a wall. The distance from the base of the wall to the base of the ladder is 6 StartRoot 2 EndRoot feet. Given this information, what can be determined about the triangle formed by the ground, wall, and ladder? Check all that apply.

The triangle is isosceles.
The leg-to-hypotenuse ratio is 1:StartRoot 2 EndRoot.
The leg-to-hypotenuse ratio is 1:StartFraction StartRoot 2 EndRoot Over 2 EndRoot.
The nonright angles are congruent.
The ladder represents the longest length in the triangle.

Answer 1

Compute the other leg: height = sqrt(12^2 - (6√2)^2) = sqrt(144 - 72) = sqrt(72) = 6√2. So the two legs are equal.

Correct choices:
- The triangle is isosceles. (Yes — the legs are both 6√2.)
- The leg-to-hypotenuse ratio is 1:√2. (Yes — 6√2 : 12 = 1 : √2.)
- The nonright angles are congruent. (Yes — an isosceles right triangle has two 45° angles.)
- The ladder represents the longest length in the triangle. (Yes — the hypotenuse is longest, 12 > 6√2.)

Incorrect:
- The leg-to-hypotenuse ratio is 1:(√2/2). (No — that would make the hypotenuse smaller than the leg; the correct form is 1:√2.)