What is the value of x? The figure shows 2 right triangles, triangle A B C with right angle B and triangle B D C with right angle D. The measure of angle B A C is 45 degrees. The measure of angle D B C is 60 degrees. The length of side C A is 6 square root 2. The length of side B D is x. Enter your answer in the box. x =
Accepted Solution
A:
Triangle ABC has: ∡ABC = 90° ∡BAC = 45° Therefore also ∡BCA = 45° This means that ABC is isosceles and that each leg is: l = √2/2 · h Since the hypotenuse of ABC is AC = 6√2 BC = √2/2 · 6√2 = 6
Now, consider the triangle BCD: ∡BDC = 90° ∡DBC = 60° which means that: ∡BCD = 30° In such triangles, the side opposite to the angle of 30° is 1/2 · h Since the hypotenuse is BC = 6, we have: BD = 1/2 · 6 = 3