In triangle ABC, the angle at vertex B is a right angle. The lengths of sides AB and BC ar… — SAT

In triangle $ABC$ , the angle at vertex $B$ is a right angle. The lengths of sides $AB$ and $BC$ are $12$ and $16$ , respectively. A point $D$ is chosen on the hypotenuse $AC$ such that $AD = 5$ . A line drawn through $D$ perpendicular to $AC$ intersects the line passing through $B$ and $C$ at point $G$ , such that $B$ lies between $G$ and $C$ . What is the length of segment $GD$ ?

Cevap: 11.25

Cevap

The length of segment

GD

is 11.25.

By the Pythagorean theorem, the hypotenuse

AC

of right triangle

ABC

\sqrt{12^2 + 16^2} = 20

. Subtracting the length of

AD

from

AC

gives

DC = 20 - 5 = 15

. Because the line

GD

is perpendicular to

AC

, the angle

\angle GDC

90^\circ

. The triangles

\triangle GDC

and

\triangle ABC

share the angle at vertex

C

and both have a right angle, which means they are similar by Angle-Angle (AA) similarity:

\triangle GDC \sim \triangle ABC

. Using the ratio of corresponding sides, we have

\frac{GD}{AB} = \frac{DC}{BC}

, which translates to

\frac{GD}{12} = \frac{15}{16}

. Solving for

GD

yields

GD = 12 \times \frac{15}{16} = 11.25

Adım Adım Çözüm

Calculate the length of the hypotenuse

AC

using the Pythagorean theorem in right triangle

ABC

AC = \sqrt{12^2 + 16^2} = 20

The length of

AC

is required to find the segment lengths on the hypotenuse.

Determine the length of segment

DC

DC = AC - AD = 20 - 5 = 15

The segment

DC

is a side of the similar triangle

\triangle GDC

that corresponds to side

BC

\triangle ABC

Establish the similarity between triangles

\triangle GDC

and

\triangle ABC

\triangle GDC \sim \triangle ABC

by AA similarity

Both triangles share the angle at vertex

C

, and both have a right angle (

\angle GDC = \angle ABC = 90^\circ

Set up the ratio of corresponding sides and solve for

GD

\frac{GD}{AB} = \frac{DC}{BC} \Rightarrow GD = 12 \times \frac{15}{16} = 11.25

The ratio of corresponding sides in similar triangles is equal.

Anahtar Kavram

Using right triangle similarity and the Pythagorean theorem to solve for unknown side lengths.

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