Part A. The given triangles are special triangles because they contain a 90° angle. So, we can use the Pythagorean theorems. However, we have to know at least two data. But since we're only given one concrete information which is the 10 units and the other being variables, we can't proceed with Pythagorean theorems. Our approach should then be ratio and proportion. As you can see, two sides of the triangle coincide at some extent. This is a manifestation that the two triangles are similar. Then. the ratio of the bases to the vertical heights are equal.
10/x = (x+7)/(x+4)
Next, we cross multiply them
10(x+4) = x(x+7)
Apply laws of algebra to solve for x.
10x + 40 = x² + 7x
x² + 7x - 10x - 40 = 0
x² - 3x - 40 = 0
To find the roots of the quadratic equation, we use the quadratic formula:
x = [-b +/- √(b² - 4ac)]/2a, where
a, b and c are the corresponding values to the general form of a quadratic equation: ax² + bx + c. So, we substitute a=1, b = -3 and c=-40.
x = [-(-3) +/- √((-3)² - 4(1)(-40))]/2(1)
x = 8, -5
Let's use the positive root as the answer. Thus x=8 units.
Part B. Now we can use the Pythagorean theorem. For ΔAED, the base is 10 units and the height is 8 units. We can then find the value of the hypotenuse:
c = √(10² + 8²)
c = 12.806 units
The perimeter is the sum of all sides.
Perimeter = 12.806 + 10 + 8
Perimeter = 30.806 units
