Question

1. in $\triangle def$ if $\overline{ef}$ is four feet longer than $\overline{de}$, $\overline{df}$ is 15 feet shorter than the sum of $\overline{ef}$ and $\overline{de}$, and the perimeter of $\triangle def$ is 88 feet, find the length of $\overline{ef}$.\

\bigcirc 32.75 ft.\
\bigcirc 27.75 ft.\
\bigcirc 29.75 ft.\
\bigcirc 31 ft.\
\bigcirc none of the above

Explanation:

Step1: Define variables for sides

Let $DE = x$, $EF = y$, $DF = z$

Step2: Translate given to equations

1. $y = x + 4$ (EF is 4ft longer than DE)
2. $z = (x + y) - 15$ (DF is 15ft shorter than sum of EF and DE)
3. $x + y + z = 88$ (Perimeter is 88ft)

Step3: Substitute y into z

Substitute $y = x+4$ into equation 2:
$z = (x + x + 4) - 15 = 2x - 11$

Step4: Substitute into perimeter equation

Substitute $y = x+4$ and $z=2x-11$ into equation 3:
$x + (x + 4) + (2x - 11) = 88$
Simplify:
$4x - 7 = 88$

Step5: Solve for x

$4x = 88 + 7 = 95$
$x = \frac{95}{4} = 23.75$

Step6: Calculate EF (y)

$y = x + 4 = 23.75 + 4 = 27.75$

Answer:

27.75 ft.