Question

restating the 45°-45°-90° triangle theorem
c = a√2
another way of stating this relationship, when youre using only one variable, is to define the leg length as __.
the ______ is defined as the length of x√2.
notice the ratio of the side lengths will still remain the same. that ratio is 1:√__

Explanation:

Step1: Analyze the 45-45-90 triangle

In a 45°-45°-90° triangle, the legs are equal (let each leg be \( x \)) and the hypotenuse \( c \) is \( x\sqrt{2} \). The first blank (leg length) should be \( x \) as the legs are defined by \( x \) here. The second blank (the side with \( x\sqrt{2} \)) is the hypotenuse. For the ratio, legs are \( x \), hypotenuse \( x\sqrt{2} \), so ratio of leg to hypotenuse (or leg to leg to hypotenuse) is \( 1:1:\sqrt{2} \), so the last blank is \( 2 \) (since \( \sqrt{2} \) has radicand 2).

Step2: Fill in the blanks

1. Leg length: When using one variable, the leg length is \( x \).
2. The side defined as \( x\sqrt{2} \) is the hypotenuse.
3. The ratio of side lengths (leg:leg:hypotenuse) is \( 1:1:\sqrt{2} \), so the radicand in the square root for the ratio part is \( 2 \).

Answer:

First blank: \( x \); Second blank: hypotenuse; Last blank: \( 2 \)

(If the blanks are in order: "x", "hypotenuse", "2")