In right triangle ABC, the measure of \angle B is 90^\circ, the measure of \angle A is 30^… — ACT

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , the measure of $\angle A$ is $30^\circ$ , and the length of leg $BC$ is $12$ . An altitude $BD$ is drawn perpendicular to the hypotenuse $AC$ . Let $E$ be the midpoint of the altitude $BD$ . A line passing through $E$ is perpendicular to $BD$ and intersects the leg $AB$ at $G$ and the leg $BC$ at $F$ . What is the length of segment $GF$ ?

Answer: 12

Answer

The length of segment

GF

12

The correct answer is

12

. By analyzing the geometric properties of the 30-60-90 right triangle

ABC

, the altitude

BD

is found to be

6\sqrt{3}

, making the half-segment

BE = 3\sqrt{3}

. The perpendicular line at

E

creates two smaller 30-60-90 right triangles,

\triangle BEF

and

\triangle BEG

. Solving for the legs along the line gives

EF = 3

and

EG = 9

, which sum to

12

Step-by-Step Solution

Find the length of altitude

BD

in right triangle

ABC

BD = 6\sqrt{3}

In right triangle

ABC

, we have

\angle B = 90^\circ

\angle A = 30^\circ

, and

\angle C = 60^\circ

. The altitude

BD

forms a smaller 30-60-90 right triangle

BCD

with hypotenuse

BC = 12

. Since

BD

is opposite the

60^\circ

angle

\angle C

, we have

BD = BC \sin(60^\circ) = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3}

Calculate the length of segment

BE

BE = 3\sqrt{3}

Since

E

is the midpoint of the altitude

BD

, we divide the length of

BD

by 2:

BE = \frac{6\sqrt{3}}{2} = 3\sqrt{3}

Determine the length of segment

EF

in right triangle

BEF

EF = 3

Since the line

GF

is perpendicular to

BD

\angle BEF = 90^\circ

. In right triangle

BCD

, we have

\angle DBC = 30^\circ

, which means

\angle EBF = 30^\circ

. This makes

\triangle BEF

a 30-60-90 right triangle where

BE

is adjacent to the

30^\circ

angle and

EF

is opposite to it. Thus,

EF = \frac{BE}{\sqrt{3}} = \frac{3\sqrt{3}}{\sqrt{3}} = 3

Determine the length of segment

EG

in right triangle

BEG

EG = 9

Since

\angle BEG = 90^\circ

and

\angle ABD = 90^\circ - \angle DBC = 60^\circ

, the angle

\angle EBG = 60^\circ

. This makes

\triangle BEG

a 30-60-90 right triangle where

BE

is adjacent to the

60^\circ

angle and

EG

is opposite to it. Thus,

EG = BE \sqrt{3} = 3\sqrt{3} \times \sqrt{3} = 9

Calculate the total length of segment

GF

GF = 12

Since

G

E

, and

F

are collinear and

E

lies between

G

and

F

, the length of segment

GF

is the sum of

EG

and

EF

GF = 9 + 3 = 12

Key Concept

Using properties of 30-60-90 special right triangles to find segment lengths in complex geometric configurations.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic