MATH SOLVE

2 months ago

Q:
# 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

Hence, x = 3

∡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

Hence, x = 3