Chapter 4.4, Problem 23PSC

To determine

To Prove: $\Delta ABC$ is equilateral.

Explanation of Solution

Given information: Altitude AD, BE and CF bisects corresponding sides.

Formula used: properties of triangle, angle sum property of triangle, congruency.

Proof:

  $\begin{array}{l}\text{AD, BE and CF}\text{are}\text{Altitude }(\text{given})\\ \text{D,E,F}\text{are}\text{midpoints}(\text{given})\\ \therefore \angle ODB=\angle ODC(\text{Each}{90}^{\circ })\\ BD=DC\\ OD=OD(\text{Reflective}\text{property)}\\ \therefore \Delta ODB\cong \Delta ODC(\text{SAS})\\ \therefore \angle OBD=\angle OCD.\\ \text{Similarly,}\angle OBF=\angle OAFand\angle OAF=\angle OAE.\\ \therefore \angle OAF+\angle OAE=\angle OCE+\angle OCD=\angle OBD+\angle OBF.\\ \therefore \angle A=\angle B=\angle C.\\ \text{Hence}\Delta ABC\text{is}\text{equilateral}\text{.}\end{array}$