Answer
$h = 2.4$ units
Work Step by Step
1. Find the area of the larger $\triangle$
Let $A = $ area of the larger $\triangle$
$A = \frac{b \times h}{2}$
$= \frac{8 \times 6}{2}$
$= \frac{48}{2}$
$= 24$ unit$^{2}$
2. Find the area of the smaller $\triangle$
Let $S =$ area of the smaller unshaded $\triangle$
$S = \frac{4 \times 3}{2}$ (We know that $A$ and $B$ are midpoints so therefore the lengths of both the base and height is halved)
$= \frac{12}{2}$
$= 6$ unit$^{2}$
3. Find the area of the trapezoid
$=(Larger \triangle)$ $- (Smaller \triangle)$
$= 24 - 6$
$= 18$ unit$^{2}$
4. Find the hypotenuse for both the small and large $\triangle$'s
4.1 Larger $\triangle$
$a^{2} + b^{2} = c^{2}$
$(8)^{2} + (6)^{2} = c^{2}$
$64 + 36 = c^{2}$
$100 = c^{2}$
$c = ±\sqrt {100}$
$c = 10$ units
4.2 Smaller $\triangle$
$a^{2} + b^{2} = c^{2}$
$(4)^{2} + (3)^{2} = c^{2}$
$16 + 9 = c^{2}$
$c^{2} = 25$
$c = ±\sqrt {25}$
$c = 5$ units
Using the formula for a trapezium, find the height of the trapezoid
Let $T =$ area of the trapezium ($a = 5$ and $b = 10$, the hypotenuse of the small and large $\triangle$'s)
$T = \frac{1}{2}(a+b)h$
$18 = (0.5)(5+10)h$
$18 = (0.5)(15)h$
$18 = \frac{15h}{2}$
$36 = 15h$
$h = 2.4$ units
