Chapter 3.8, Problem 8PSB

To determine

To prove : $\Delta ACF$ is isosceles.

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{BD}\perp \overrightarrow{CF}$

  $\overrightarrow{GE}\perp \overrightarrow{CF}$

  $\overrightarrow{CE}\cong \overrightarrow{DF}$

  $\overrightarrow{BC}\cong \overrightarrow{GF}$

Formula used : The similar angles of congruent triangles are congruent

Proof : In triangle BCD and GFE, we can say that

  $\angle BDC\cong \angle GEF={90}^0$

  $\overrightarrow{CD}\cong \overrightarrow{EF}$ ( by using $\overrightarrow{CE}\cong \overrightarrow{DF}$ )

  $\overrightarrow{BC}\cong \overrightarrow{GF}$

Thus, we can say that $\Delta BCD\cong \Delta GFE$ by SAS property

Now, as we know that similar angles of congruent triangles are congruent, we can say that

  $\angle BCD\cong \angle GFE$

Thus, we can also say that

  $\angle ACF\cong \angle AFC$

Hence, proved that $\Delta ACF$ is isosceles