A right triangle has a hypotenuse of length x + 4 inches. The lengths of the two legs of t… — ACT

A right triangle has a hypotenuse of length $x + 4$ inches. The lengths of the two legs of the triangle are $x$ inches and $x + 2$ inches. What is the value of $x$ ?

A
2
6Answer
C
8
D
10
E
12

Answer

The correct answer is 6 because applying the Pythagorean theorem yields the relation

x^2 + (x + 2)^2 = (x + 4)^2

. Expanding the binomials gives

x^2 + x^2 + 4x + 4 = x^2 + 8x + 16

, which simplifies to the quadratic equation

x^2 - 4x - 12 = 0

. Factoring this equation yields

(x - 6)(x + 2) = 0

. Discarding the negative solution

x = -2

because length must be positive leaves the correct solution of 6.

Step-by-Step Solution

Set up the equation using the Pythagorean theorem, where the sum of the squares of the legs equals the square of the hypotenuse.

x^2 + (x + 2)^2 = (x + 4)^2

The sides of a right triangle must satisfy the Pythagorean relation

a^2 + b^2 = c^2

Expand the squared binomial terms on both sides of the equation.

x^2 + (x^2 + 4x + 4) = x^2 + 8x + 16

Applying the algebraic identity

(a+b)^2 = a^2 + 2ab + b^2

is necessary to simplify the terms.

Simplify the equation and move all terms to one side to set the quadratic expression to zero.

x^2 - 4x - 12 = 0

Standard form (

ax^2 + bx + c = 0

) is required to solve quadratic equations.

Factor the quadratic equation.

(x - 6)(x + 2) = 0

Factoring allows finding the roots by setting each linear binomial factor to zero.

Solve for

x

and discard any physically impossible negative values.

x = 6

(since

x = -2

is discarded)

A physical measurement of side length must be strictly positive.

Key Concept

Formulating and solving quadratic equations derived from the Pythagorean theorem by expanding binomials and factoring.

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