Question 304·Medium·Math

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In the figure above, segment $DE$ is drawn parallel to side $\overline{BC}$ of $\triangle ABC$ . If $AD = 4.8$ , $DB = 3.2$ , and $AE = 5.4$ , what is the length of $\overline{AC}$ ?

A segment parallel to a triangle’s side creates similar triangles: the smaller triangle and the whole triangle have matching angles, so their corresponding sides have the same ratio. In a nested figure, compare whole sides, not a side with a leftover piece. Once you’ve matched the sides, you can solve a proportion with decimals in Desmos.

Hints

Hint 1
The parallel segment makes the angle at $D$ equal the angle at $B$ . The triangles also share the angle at $A$ . Those two equal angle pairs make the triangles similar. Which vertices match?
Hint 2
A corresponding side plays the same role in each triangle. The small triangle uses $AD$ ; the large one uses all of $AB$ , not only $DB$ . How can you find $AB$ ?
Hint 3
The pair $AD,AB$ has the same small-to-large ratio as $AE,AC$ . Write both ratios in that order, then solve for the length the question asks for.

Step-by-step

Match the triangles and solve a proportion
Step 1
Identify the matching triangles
Because $DE$ is parallel to $BC$ , $\angle ADE=\angle ABC$ . Both triangles also have the angle at $A$ . Two pairs of equal angles give AA similarity, meaning the triangles have the same shape, so $\triangle ADE\sim\triangle ABC$ . The matches are $A\leftrightarrow A$ , $D\leftrightarrow B$ , and $E\leftrightarrow C$ .
Step 2
Find the whole side of the large triangle
$D$ lies between $A$ and $B$ , so add the two pieces to get the whole side:
$AB=AD+DB$ $AB=4.8+3.2$ $AB=8.$
Step 3
Match sides and write the proportion
The vertex matches pair $AD$ with $AB$ , and $AE$ with $AC$ . Use the whole side $AB$ , not the leftover $DB$ . Corresponding sides of similar triangles have the same ratio, so
$\frac{AD}{AB}=\frac{AE}{AC}.$
Step 4
Solve for the requested length
Let $x_1$ mean $AC$ . Type $\frac{4.8}{8}\sim\frac{5.4}{x_1}$ in Desmos. The $\sim$ tells Desmos to find the value of $x_1$ that makes the ratios equal. Under PARAMETERS, it shows $x_1=9$ , so $AC=9$ . The length of $\overline{AC}$ is $9$ . Choice B.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreSolve similar-triangle proportions

Question 304·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

Hints

Step-by-step

Lessons that teach this