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
6Cevap
C
8
D
10
E
12

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

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