Tromika wants to calculate the area of the isosceles triangle at right. 

1. She decides to start by drawing a height from vertex $A$ to side $\overline{BC}$ as shown below. Will the two smaller triangles be congruent? In other words, is $∆ABD ≅ ∆ACD$? Why or why not?

   

   Hint (a):

   Look at the two triangles that are created. They share a side, and the tick marks mean the side lengths are equal. Are the triangles congruent?

   Answer (a):

   Yes, they are congruent by $\text{HL}≅$.

2. What is $m∠DAB$? $BD$?

   Hint (b):

   $∠DAB≅∠DAC$ since congruent triangles have congruent parts. 

   Answer (b):

   $m∠DAB=18^{\circ}$, $BD = 4$

3. Determine $AD$. Show how you got your answer.

   Hint (c):

   Which trig ratio can you use to determine this length?

4. Calculate the area of $∆ABC$.

   Hint (d):

   How do you find the area of a triangle? Do you have enough information to calculate the area?