In triangle ABC, the measure of angle ABC is 90^\circ and the measure of angle BAC is 30^\… — SAT

In triangle $ABC$ , the measure of angle $ABC$ is $90^\circ$ and the measure of angle $BAC$ is $30^\circ$ . Segment $BD$ is perpendicular to segment $AC$ such that $D$ lies on $AC$ . Segment $DE$ is perpendicular to segment $BC$ such that $E$ lies on $BC$ . If the length of segment $CE$ is $3$ , what is the length of segment $AC$ ?

Answer: 24

Answer

The correct answer is 24.

The correct answer is 24. Since the question asks for the length of segment AC, and by sequentially applying the properties of 30-60-90 right triangles we find CD = 6, BC = 12, and AC = 24.

Step-by-Step Solution

Find the measure of angle

ACB

in triangle

ABC

\angle ACB = 60^\circ

The sum of the angles in a triangle is

180^\circ

, so

\angle ACB = 180^\circ - \angle ABC - \angle BAC = 180^\circ - 90^\circ - 30^\circ = 60^\circ

Calculate the length of segment

CD

using right triangle

DEC

CD = 6

In the right triangle

DEC

\angle DEC = 90^\circ

and

\angle C = 60^\circ

, making it a 30-60-90 triangle. The side opposite the

30^\circ

angle is

CE = 3

, so the hypotenuse

CD

2 \times CE = 2 \times 3 = 6

Calculate the length of segment

BC

using right triangle

BDC

BC = 12

In the right triangle

BDC

BD

is perpendicular to

AC

, so

\angle BDC = 90^\circ

. With

\angle BCD = 60^\circ

, this is a 30-60-90 triangle. The side opposite the

30^\circ

angle is

CD = 6

, so the hypotenuse

BC

2 \times CD = 2 \times 6 = 12

Calculate the length of the hypotenuse

AC

using right triangle

ABC

AC = 24

In the right triangle

ABC

, the angle

\angle BAC = 30^\circ

and the side opposite to it is

BC = 12

. The hypotenuse

AC

is twice the length of the opposite leg, so

AC = 2 \times BC = 2 \times 12 = 24

Key Concept

Properties of special right triangles (30-60-90 triangles) and their trigonometric ratios.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic