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 greater than or equal to 13 squared

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

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

x2+(x+5)2≤132

Answer 1

To find the quadratic inequality that represents the length of the shorter leg \( x \), we first define the lengths of the legs of the right triangle.

Let:
- The shorter leg be \( x \) cm.
- The longer leg be \( x + 5 \) cm (since it is 5 cm longer than the shorter leg).

According to the Pythagorean theorem, the relationship between the lengths of the legs and the hypotenuse in a right triangle is given by:

\[
a^2 + b^2 = c^2
\]

Where:
- \( a \) and \( b \) are the lengths of the legs,
- \( c \) is the length of the hypotenuse.

In this case, we have:

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

Thus, we can expand and combine the expressions:

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

This simplifies to:

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

Rearranging gives:

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

Now, rearranging the terms, we rewrite this as:

\[
x^2 + 5x - 72 \geq 0
\]

However, the correct format of the inequality only represents the lengths derived from the Pythagorean theorem at a minimum hypotenuse of 13 cm.

Thus, the correct option that represents this inequality is:

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

So the correct choice is:

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