Question

In parallelogram $ABCD,$ $\overline{DE}$ is the altitude to the base $\overline{AB}$ and $\overline{DF}$ is the altitude to the base $\overline{BC}.$ (Both pictures represent the same parallelogram.) If $DC = 12,$ $EB = 4,$ and $DE = 6,$ then $DF =$

$6.4$

$7$

$7.2$

$8$

$10$

Answer: C

Concepts:parallelogram Pythagorean Theorem

Difficulty rating: 1150

Small Hint:

Opposite sides are equal, so $AB = 12;$ find $AE,$ then $AD$ with the right triangle $ADE$

Big Hint:

The area equals base $\times$ height for either base: $AB \cdot DE = BC \cdot DF$

Solution:

Since $AB = DC = 12,$ we get $AE = 12 - 4 = 8.$ In right triangle $ADE,$ $AD = \sqrt{8^2 + 6^2} = 10,$ so $BC = AD = 10.$

The area is $AB \cdot DE = 12 \cdot 6 = 72,$ and also $BC \cdot DF = 10 \cdot DF.$ So $DF = \dfrac{72}{10} = 7.2.$

Thus, the correct answer is C .

1995 AMC 8 Problem 24

Answer: C

Difficulty rating: 1150

Small Hint:

Big Hint:

Solution:

Problem 24 in Other Years