Answer:
The length EI is;
[tex]EI=19[/tex]
The measure of angle IFE is;
[tex]m\angle IFE=37^{\circ}[/tex]
Explanation:
Given the rectangle in the attached image.
Given;
[tex]\begin{gathered} FH=4x-2 \\ EG=5x-12 \\ m\angle IGF=53^{\circ} \end{gathered}[/tex]
Recall that the length of the diagonals of a rectangle are equal so;
[tex]\begin{gathered} FH=EG \\ 4x-2=5x-12 \end{gathered}[/tex]
solving for x, we have;
[tex]\begin{gathered} 4x-2=5x-12 \\ 12-2=5x-4x \\ x=10 \end{gathered}[/tex]
Since we have the value of x, let us substitute to get the length of diagonal EG;
[tex]\begin{gathered} EG=5x-12 \\ EG=5(10)-12=50-12 \\ EG=38\text{ units} \end{gathered}[/tex]
Also, note that the diagonals of a rectangle bisect each other, so the length of EI would be;
[tex]\begin{gathered} EI=\frac{EG}{2}=\frac{38}{2} \\ EI=19 \end{gathered}[/tex]
Therefore, the length EI is;
[tex]EI=19[/tex]
To get the measure of angle IFE;
[tex]m\angle IGF=m\angle IFG=53^{\circ}[/tex]
Reason: base angles of an isosceles triangle are equal.
So;
[tex]m\angle IFE+m\angle IFG=90^{\circ}[/tex]
Reason: Complementary angles.
Substituting the value of angle IFG;
[tex]\begin{gathered} m\angle IFE+53^{\circ}=90^{\circ} \\ m\angle IFE=90^{\circ}-53^{\circ} \\ m\angle IFE=37^{\circ} \end{gathered}[/tex]
Therefore, the measure of angle IFE is;
[tex]m\angle IFE=37^{\circ}[/tex]
