Question

the hypotenuse of a right triangle is 34 inches. one leg of the triangle is 14 inches less than the other leg. in simplified form, which equation could be used to find the lengths of the legs? a $2x^2 - 28x = 960$ b $2x^2 - 28x = 1176$ c $2x^2 - 14x = 34$ d $x^2 + 14x = 960$

Explanation:

Step1: Define variables

Let one leg be \( x \) inches. Then the other leg is \( x - 14 \) inches (since one leg is 14 inches less than the other). The hypotenuse is 34 inches.

Step2: Apply Pythagorean theorem

The Pythagorean theorem states that for a right triangle, \( a^{2}+b^{2}=c^{2} \), where \( a \) and \( b \) are the legs and \( c \) is the hypotenuse. Substituting the values, we get \( x^{2}+(x - 14)^{2}=34^{2} \).

Step3: Expand the equation

Expand \( (x - 14)^{2} \): \( (x - 14)^{2}=x^{2}-28x + 196 \). So the equation becomes \( x^{2}+x^{2}-28x + 196 = 1156 \).

Step4: Simplify the equation

Combine like terms: \( 2x^{2}-28x+196 = 1156 \). Subtract 1156 from both sides: \( 2x^{2}-28x+196 - 1156=0 \), which simplifies to \( 2x^{2}-28x - 960 = 0 \). Divide both sides by 1 (or we can also write it as \( 2x^{2}-28x=960 \)) by adding 960 to both sides of \( 2x^{2}-28x - 960 = 0 \).

Answer:

A. \( 2x^{2}-28x = 960 \)