In triangle ABC, point D lies on side BC such that the ratio of BD to DC is 1 to 2. Point … — SAT

In triangle $ABC$ , point $D$ lies on side $BC$ such that the ratio of $BD$ to $DC$ is $1$ to $2$ . Point $E$ lies on segment $AD$ such that the ratio of $AE$ to $ED$ is $3$ to $1$ . A line passing through $E$ and parallel to $AC$ intersects side $AB$ at $F$ and side $BC$ at $G$ . If the area of triangle $ABC$ is $128$ , what is the area of triangle $BFG$ ?

Answer: 32

Answer

The correct answer is 32. By establishing a coordinate system, we find that the similarity ratio of triangle

BFG

to triangle

ABC

is exactly

1/2

regardless of the triangle's shape. Since the ratio of the areas of similar triangles is the square of the similarity ratio, the area of triangle

BFG

(1/2)^2 = 1/4

of the area of triangle

ABC

, which is

128 \times 1/4 = 32

Step-by-Step Solution

Set up a coordinate system to represent the triangle's vertices.

Let

B = (0, 0)

and

C = (3, 0)

. Since

D

lies on side

BC

and

BD:DC = 1:2

, the coordinates of

D

are

(1, 0)

. Let

A = (a, b)

Setting up coordinates simplifies the proof by allowing algebraic verification of the ratio.

Calculate the coordinates of point

E

on segment

AD

Using the section formula with ratio

AE:ED = 3:1

E = \left(\frac{1(a) + 3(1)}{4}, \frac{1(b) + 3(0)}{4}\right) = \left(\frac{a+3}{4}, \frac{b}{4}\right)

Finding the coordinates of

E

is necessary to determine the equation of line

FG

Find the equation of line

FG

which is parallel to

AC

and passes through

E

The slope of

AC

m = \frac{-b}{3-a}

. The equation of

FG

y - \frac{b}{4} = \frac{-b}{3-a}\left(x - \frac{a+3}{4}\right)

The line

FG

is parallel to

AC

, meaning they share the same slope.

Find the coordinates of

G

by setting

y = 0

in the equation of

FG

Setting

y = 0

yields

-\frac{b}{4} = \frac{-b}{3-a}\left(x_G - \frac{a+3}{4}\right) \implies x_G = 1.5

. Thus,

G = (1.5, 0)

Point

G

is the intersection of the line

FG

with side

BC

(the x-axis).

Determine the similarity ratio and calculate the area of triangle

BFG

Since

G

is the midpoint of

BC

, the similarity ratio of

\triangle BFG

\triangle BAC

k = \frac{1}{2}

. The ratio of their areas is

k^2 = \frac{1}{4}

. The area of

\triangle BFG

128 \times \frac{1}{4} = 32

Similar triangles have area ratios equal to the square of their similarity ratio.

Key Concept

The ratio of the areas of similar triangles is equal to the square of their similarity ratio.

Alternative Method

Using Menelaus's Theorem on triangle

ABD

and transversal

FEG

can also establish the midpoint relations directly without using coordinates.

Estimated Time:3m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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