Question

In this diagram $AB$ and $AC$ are the equal sides of an isosceles $△ABC$, in which is inscribed equilateral triangle $DEF$. Designate $\angle BFD$ by $a$, $\angle ADE$ by $b$, and $\angle FEC$ by $c$. Then:

• A. $b=\frac{a+c}{2}$
• B. $b=\frac{a-c}{2}$
• C. $a=\frac{b-c}{2}$
• D. $a=\frac{b+c}{2}$
• E. None of these

Answer 1

Answer: (E)

None of these

ABC is an isosceles triangle.

$\therefore \angle B=\angle C$

DEF is an isosceles triangle.

$\therefore \angle EDF=\angle DFE=\angle FED=60°\therefore \angle BDF=180°-(60°+b)=180°-b\therefore \angle EFC=180°-(60°+a)=180°-a\therefore \angle B=180°-(\angle BDF+\angle BFD)=180°-(120°-b+a)=60°+b-a\therefore \angle C=180°-(\angle CFE+\angle CEF)=180°-(120°-a+c)=60°+a-c\therefore 60°+b-a=60°+a-c[\because \angle B=\angle C]⟹2a=b+c⟹a=\frac{b+c}{2}$