Question

What is the value of x? Enter your answer in the box. x =

Answer 1

The triangle on the left is a 45-45-90 triangle so if its hypotenuse is 6√2, the legs are each 6 units long. 

As such the hypotenuse of the right triangle is 6 units. The right triangle is a 30-60-90 triangle. So your x leg is half the hypotenuse. 6 / 2 = 3

x = 3 

Answer 2

Use the special properties of the right triangles in this diagram.

Triangle ABC has defined slopes of 90 degrees at angle ABC, and 45 degrees at angle BAC. Therefore, angle ACB will be 45 degrees, as the slopes in a triangle must all add up to 180. This makes the triangle a 45-45-90 triangle.

A 45-45-90 triangle has the same proportions for its side lengths. The opposite and adjacent side lengths will be x, and the hypotenuse's side length will be x√2. 

The hypotenuse's side length is given as 6√2. To find the other side lengths in this triangle, remove the square root. Sides AB and BC both have a length of 6.

Triangle BCD has defined angles of 90 degrees for angle CDB, and 60 degrees for angle DBC. Therefore, angle BCD will have an angle of 30 degrees, so that all the angles in this triangle add up to 180. This makes the triangle a 30-60-90 triangle.

A 30-60-90 triangle also has proportional side lengths. The adjacent side's length will be x, the opposite side's length will be x√3, and the hypotenuse's side length will be 2x.

Hypotenuse BC already has a side length of 6. Therefore, 2x will equal 6. We are looking for the opposite side BD's length, so we must solve for x:

$2x = 6$

Divide both sides by 2 to get x by itself:

$x = 3$

Side BD's length is 3.