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

The length EI is;

`EI=19`

The measure of angle IFE is;

`m\angle IFE=37^(\circ)`

Step-by-step 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=(EG)/(2)=(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)`