Answer:
A ≈ 14.079 square units
Step-by-step explanation:
Area of a triangle is one half the base times the height.
A = ½ bh
A = ½ (10) (2x)
A = 10x
We need to find the value of x.
Starting with the triangle on the left, use Pythagorean theorem to find the length of the base.
(3x)² = (2x)² + a²
9x² = 4x² + a²
a² = 5x²
a = x√5
Repeat for the triangle on the right:
(x + 6)² = (2x)² + b²
x² + 12x + 36 = 4x² + b²
b² = -3x² + 12x + 36
The two bases add up to 10:
a + b = 10
Subtract a from both sides, then square both sides:
b = 10 − a
b² = 100 − 20a + a²
Substitute and simplify:
-3x² + 12x + 36 = 100 − 20(x√5) + 5x²
0 = 64 − (12 + 20√5) x + 8x²
0 = 2x² − (3 + 5√5) x + 16
Solve with quadratic formula:
x = [ -b ± √(b² − 4ac) ] / 2a
x = [ (3 + 5√5) ± √((-(3 + 5√5))² − 4(2)(16)) ] / 2(2)
x = [ (3 + 5√5) ± √(9 + 30√5 + 125 − 128) ] / 4
x = [ (3 + 5√5) ± √(6 + 30√5) ] / 4
x ≈ 1.4079, 5.6823
If we substitute 5.6823 into our a and b equations, we find that a = 12.706 and b = 7.322, which add up to 20.028, not 10.
So x ≈ 1.4079.
Therefore the area is:
A ≈ 14.079
