Question

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

$b=\dfrac{a+c}{2}$

$b=\dfrac{a-c}{2}$

$a=\dfrac{b-c}{2}$

$a=\dfrac{b+c}{2}$

none of these

Answer: D

Concepts:angle chasing isosceles triangle equilateral triangle

Difficulty rating: 2000

Small Hint:

Let each base angle of $ABC$ be $\theta$

Big Hint:

Compare the directions of $DF,$ $DE,$ and $EF$ using the $60^\circ$ angles of the equilateral triangle

Solution:

Let each base angle of $ABC$ be $\theta.$ Measured from the direction $BC,$ the line $FD$ has direction $180^\circ-a,$ so the equilateral condition makes $DE$ have direction $60^\circ-a.$ At $D,$ $b=\theta-(60^\circ-a)=\theta-60^\circ+a.$ Similarly, angle chasing at $E$ gives $c=60^\circ+a-\theta.$ Adding these equations yields $b+c=2a.$

Therefore $a=\frac{b+c}{2},$ and the correct answer is D.

1960 AMC 12 Problem 38

Answer: D

Difficulty rating: 2000

Small Hint:

Big Hint:

Solution:

Problem 38 in Other Years