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PLEASE HELP 15 POINTS

What is the value of x?

Enter your answer, as an exact value, in the box.

PLEASE HELP 15 POINTS What is the value of x? Enter your answer, as an exact value-example-1

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x = 3√2.

The top triangle has one angle marked 30 and one marked 90; this makes the other triangle 180-(30+90) = 180-120 = 60.

In a 30-60-90 triangle, the sides are given by t, t√3, and 2t; t is across from the 30° angle; t√3 is across from the 60° angle; and 2t is across from the 90° angle. The side marked 6 is across from the 90° angle, so

2t = 6
t = 3

This means that the side between the triangles is 3.

Since the bottom triangle has one angle marked 45 and another marked 90, the third angle would be 180 - (45+90) = 180 - 135 = 45°. In a 45-45-90 triangle, the sides are listed as t, t, and t√2; the two sides marked t are across from the 45° angles and the side marked t√2 is across from the 90° angle. This means t = 3, so x = t√2 = 3√2.
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