Answer:
- x = 8 +8√3 ≈ 21.856
- y = 12 +4√3 ≈ 18.928
Explanation:
Given a diagram of two "special" triangles, you want to know the measures of x and y.
Special triangles
The right triangles with angles 30°-60°-90° and 45°-45°-90° are considered "special" because they have side lengths in ratios that can be expressed in a simple form.
The side lengths of a 30°-60°-90° triangle have ratios 1 : √3 : 2.
The side lengths of a 45°-45°-90° triangle have ratios 1 : 1 : √2.
Application
The ratio of x to y is the ratio of the two longer sides of the 30°-60°-90° triangle:
y : x = √3 : 2
The unmarked segment at the bottom edge of the figure will have length x/2. The ratio of side lengths of the 45°-45°-90° triangle is then ...
1 : 1 = (8 +x/2) : y
These relations give us two equations in x and y:
Solution
Substituting the latter expression for y into the first equation gives ...
2(8 +x/2) = (√3)x . . . . . . substitute for y
16 +x = √3x . . . . . . . . . . eliminate parentheses
16 = x(√3 -1) . . . . . . . . . subtract x and factor
x = 16/(√3 -1) . . . . . . . . divide by the coefficient of x
x = 16(√3 +1)/2 = 8(√3 +1)
x = 8 +8√3 ≈ 21.856
Then the value of y is ...
y = 8 +(8 +8√3)/2 = 8 +4 +4√3
y = 12 +4√3 ≈ 18.928