QuestionA right triangle has a shorter leg $x$ , a longer leg $x + 7$ , and a hypotenuse $x + 9$ . Find $x$ .

Studdy Solution

STEP 1

Assumptions1. The longer leg of the triangle is7 cm longer than the shorter leg. . The hypotenuse is9 cm longer than the shorter leg.
3. The triangle is a right triangle, so the Pythagorean theorem applies.

STEP 2

Let's denote the length of the shorter leg as $x$ . Then, the longer leg is $x +7$ and the hypotenuse is $x +9$ .

STEP 3

We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. $(x +7)^2 + x^2 = (x +9)^2$

STEP 4

Expand the squares on both sides of the equation.
$x^2 +14x +49 + x^2 = x^2 +18x +81$

STEP 5

Combine like terms.
$2x^2 +14x +49 = x^2 +18x +81$

STEP 6

Subtract $x^2 +18x +81$ from both sides to set the equation to zero.
$x^2 -4x -32 =0$

STEP 7

Factor the quadratic equation.
$(x -)(x +4) =0$

STEP 8

Set each factor equal to zero and solve for $x$ .
$x -8 =0 \Rightarrow x =8$ $x +4 =0 \Rightarrow x = -4$

STEP 9

We discard $x = -4$ because lengths cannot be negative. So, the length of the shorter leg is8 cm.

STEP 10

Substitute $x =8$ into the expressions for the longer leg and the hypotenuse.
$Longer\, leg = x +7 =8 +7 =15\, cm$ $Hypotenuse = x +9 =8 +9 =17\, cm$ Length of the shorter leg8 cmLength of the longer leg15 cmLength of the hypotenuse17 cm

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