In the xy-plane, triangle ABC is similar to triangle DEF, where vertices A, B, and C corre… — SAT

In the $xy$ -plane, triangle $ABC$ is similar to triangle $DEF$ , where vertices $A$ , $B$ , and $C$ correspond to vertices $D$ , $E$ , and $F$ , respectively. The vertices of triangle $ABC$ are $A(0, 0)$ , $B(6, 0)$ , and $C(0, 8)$ . The vertices of triangle $DEF$ are $D(2, 1)$ , $E(11, 1)$ , and $F(2, y)$ , where $y > 1$ . What is the value of $y$ ?

A
7
B
12
13Answer
D
16

Answer

The correct answer is 13. Since triangle

ABC

is similar to triangle

DEF

, the ratio of their corresponding sides is constant. Side

AB

has length

6

, and the corresponding side

DE

has length

11 - 2 = 9

. This gives a scale factor of

\frac{9}{6} = 1.5

. Applying this scale factor to the vertical side

AC

(length

8

) yields a length of

12

for the corresponding side

DF

. Since

D

has coordinates

(2, 1)

and

F

has coordinates

(2, y)

with

y > 1

, the length of

DF

y - 1 = 12

, which solves to

y = 13

Step-by-Step Solution

Determine the lengths of the corresponding sides of triangle

ABC

The vertices of triangle

ABC

are

A(0, 0)

B(6, 0)

, and

C(0, 8)

. The length of horizontal side

AB

6 - 0 = 6

, and the length of vertical side

AC

8 - 0 = 8

To find the similarity ratio between the two triangles, we need the lengths of the sides of the first triangle.

Determine the length of the corresponding side

DE

of triangle

DEF

The vertices of side

DE

are

D(2, 1)

and

E(11, 1)

. Since they share the same

y

-coordinate, this is a horizontal segment with a length of

11 - 2 = 9

Since vertex

D

corresponds to

A

and vertex

E

corresponds to

B

, side

DE

corresponds to side

AB

Calculate the scale factor between the similar triangles.

The scale factor from triangle

ABC

to triangle

DEF

\frac{DE}{AB} = \frac{9}{6} = 1.5

Similar triangles have corresponding side lengths that are proportional.

Find the length of side

DF

and the coordinate

y

Side

DF

corresponds to side

AC

. The length of

DF

1.5 \times AC = 1.5 \times 8 = 12

. Since

D

is at

(2, 1)

and

F

is at

(2, y)

, the vertical distance is

y - 1 = 12

, which gives

y = 13

We apply the scale factor to the corresponding side length and use the coordinate of

D

to solve for the coordinate

y

F

Key Concept

Using the properties of similar triangles and coordinates in the plane to solve for unknown lengths and coordinates.

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