Question

One leg of a right triangle is 5 centimeters longer than the other leg. What quadratic inequality represents the length of the shorter leg, x, when the hypotenuse is at least 13 centimeters long?(1 point)
Responses

x2+(5x)2≤132
x squared plus 5 x squared is less than or equal to 13 squared

x2+(x+5)2≤132
x squared plus open paren x plus 5 close paren squared is less than or equal to 13 squared

x2+(5x)2≥132
x squared plus 5 x squared is greater than or equal to 13 squared

x2+(x+5)2≥132

Answer 1

To solve this problem, we first define the legs of the right triangle. Let \( x \) be the length of the shorter leg. Since one leg is 5 centimeters longer than the other, the other leg can be described as \( x + 5 \).

According to the Pythagorean theorem, the relationship between the legs and the hypotenuse \( c \) of a right triangle is given by:

\[
x^2 + (x + 5)^2 = c^2
\]

In this case, we want the hypotenuse to be at least 13 centimeters long, which means:

\[
c \geq 13
\]

Thus, we need to ensure that:

\[
x^2 + (x + 5)^2 \geq 13^2
\]

Calculating \( 13^2 \):

\[
13^2 = 169
\]

Substituting this into our inequality gives us:

\[
x^2 + (x + 5)^2 \geq 169
\]

Now we simplify \( (x + 5)^2 \):

\[
(x + 5)^2 = x^2 + 10x + 25
\]

Therefore, our inequality becomes:

\[
x^2 + x^2 + 10x + 25 \geq 169
\]

Combining like terms results in:

\[
2x^2 + 10x + 25 \geq 169
\]

Subtracting 169 from both sides gives:

\[
2x^2 + 10x - 144 \geq 0
\]

To match this with the original responses, we will analyze the provided options. The correct quadratic inequality representing the length of the shorter leg \( x \) is:

\[
x^2 + (x + 5)^2 \geq 13^2
\]

Thus, the correct response is:

**\( x^2 + (x + 5)^2 \geq 13^2 \)**.