Question

in △abc ~ △def, if ab = 6, de = 9, and ac = 8, what is the length of df?
a. 18
b. 10
c. 12
d. 16

in △abc, d and e are midpoints of ab and ac, respectively. prove that (overline{de}paralleloverline{bc}). fill in the missing reason.

statement	reason
de=(\frac{1}{2}bc)	definition of midsegment
(overline{de}paralleloverline{bc})	____

a. midsegment theorem
b. alternate interior angles theorem
c. csstp
d. aa similarity criterion

Explanation:

First sub - question

Step1: Use property of similar triangles

For similar triangles $\triangle ABC\sim\triangle DEF$, the ratios of corresponding sides are equal. That is $\frac{AB}{DE}=\frac{AC}{DF}$. We are given $AB = 6$, $DE=9$, and $AC = 8$. Let the length of $DF$ be $x$. Then $\frac{6}{9}=\frac{8}{x}$.

Step2: Cross - multiply

Cross - multiplying the proportion $\frac{6}{9}=\frac{8}{x}$ gives us $6x=9\times8$.

Step3: Solve for $x$

$6x = 72$, so $x=\frac{72}{6}=12$.

The midsegment theorem states that the midsegment of a triangle (a line segment joining the midpoints of two sides of a triangle) is parallel to the third side and half its length. Since we have already established that $DE=\frac{1}{2}BC$ based on the definition of midsegment and we want to prove $DE\parallel BC$, the reason is the Midsegment Theorem.

Answer:

c. 12

Second sub - question