Question

In the rectangle below, FH = 4x – 2, EG= 5x-12, and m ZIGF = 53º.Find El and m ZIFE.EFBEI =Хm LIFE =HG

Answer 1

Answer:

The length EI is;

$EI=19$

The measure of angle IFE is;

$m\angle IFE=37^{\circ}$

Explanation:

Given the rectangle in the attached image.

Given;

$\begin{gathered} FH=4x-2 \\ EG=5x-12 \\ m\angle IGF=53^{\circ} \end{gathered}$

Recall that the length of the diagonals of a rectangle are equal so;

$\begin{gathered} FH=EG \\ 4x-2=5x-12 \end{gathered}$

solving for x, we have;

$\begin{gathered} 4x-2=5x-12 \\ 12-2=5x-4x \\ x=10 \end{gathered}$

Since we have the value of x, let us substitute to get the length of diagonal EG;

$\begin{gathered} EG=5x-12 \\ EG=5(10)-12=50-12 \\ EG=38\text{ units} \end{gathered}$

Also, note that the diagonals of a rectangle bisect each other, so the length of EI would be;

$\begin{gathered} EI=\frac{EG}{2}=\frac{38}{2} \\ EI=19 \end{gathered}$

Therefore, the length EI is;

$EI=19$

To get the measure of angle IFE;

$m\angle IGF=m\angle IFG=53^{\circ}$

Reason: base angles of an isosceles triangle are equal.

So;

$m\angle IFE+m\angle IFG=90^{\circ}$

Reason: Complementary angles.

Substituting the value of angle IFG;

$\begin{gathered} m\angle IFE+53^{\circ}=90^{\circ} \\ m\angle IFE=90^{\circ}-53^{\circ} \\ m\angle IFE=37^{\circ} \end{gathered}$

Therefore, the measure of angle IFE is;

$m\angle IFE=37^{\circ}$