We are asked to solve the perimeter of the given triangle QSU. To solve this, we need to formulate two equations, please refer to the attached picture.
For triangle SRC, we have it:
(2x)² + r² = h²
For triangle STC, we have it:
(x+3)² + r² = h²
Perform subtraction of two equations:
(2x)² + r² = h²
- ((x+3)² + r² = h²)
---------------------------
(2x)² - (x+3)² = 0
Simplify further the expression such as:
4x² - x² - 6x - 9 =0
3x² - 6x - 9 =0
(3x² - 6x - 9 =0 ) / 3
Performing quadratic equation, then roots are:
x² - 2x - 3 = 0
(x-3) (x + 1) = 0
x1 = 3
x2 = -1 (not possible)
Using x=3, the total perimeter for two sides of the triangle is:
Partial perimeter = 2x + (x+3) + 10 + 4
Partial perimeter = 2*3+ (3+3) + 10 + 4
Partial perimeter = 26 units
Then the answer is 40 units for the total perimeter of the triangle QSU since it is impossible that the last and third side is 4 units, then select 40 units for the total perimeter.