In trapezoid $EFGH,$ $\overline{EF} \parallel \overline{GH}$. Let $P$ be the midpoint of $\overline{EH}$. If the area of triangle $EFP$ is $8$, and the area of triangle $GHP$ is $21$, then find the area of trapezoid $EFGH$.

58

Step-by-step explanation:

Given that in trapezoid EFGH, (picture enclosed)

EF is parallel to GH.

P is the mid point of non parallel side EH, let Q be mid point of non parallel side FG

Since EP = PH and FQ = QG

by parallel lines when make equal intercepts we get

PQ = 1/2 (EF+FG)

Area of trapezium = 1/2 (EF+FG)*h where h is the altitude

We find that height for both triangles EFP and GHP would be the same from equal intercepts theorem for any transversal.

So we get 1/2 * EF*h/2 = 8 and 1/2 *GH *h/2 = 21

i.e. (EF+FG)h/4 =29

So area = 1/2 (EF+FG)h = 58

f

step-by-step explanation:

f

m∠cdf = 43° , m∠gdf = 114°, m∠gde = 23°, m∠edh = 23° , m∠cdh = 157°

step-by-step explanation:

since, it is given that de bisects the angle gdh ⇒ de divides the angle gdh into two equal angles.

⇒ m∠gde = m∠edh

⇒ 8·x   - 1 = 6·x + 5

⇒ x = 3

so, m∠gde = 8·x - 1 = 23°

and m∠edh = 23°

now, it is given ∠cde is an straight angle so, m∠cde = 180°

⇒ m∠cdf + m∠fdg + m∠gde = 180°

⇒ 43° + m∠fdg + 23° = 180°

⇒ m∠fdg = 114°

again, m∠cde = m∠cdh + m∠edh = 180°

m∠cdh + 23° = 180°

⇒ m∠cdh = 157°