A surveyor is measuring a triangular plot of land, ABC. The distance from point A to point… — ACT

A surveyor is measuring a triangular plot of land, $ABC$ . The distance from point $A$ to point $C$ is $12$ meters, and the distance from point $B$ to point $C$ is $6\sqrt{2}$ meters. If the measure of angle $BAC$ is $30^\circ$ , what is the measure, in degrees, of the acute angle $ABC$ ?

Answer: 45 degrees

Answer

The measure of the acute angle

ABC

45

degrees.

Applying the Law of Sines yields the relation

\frac{\sin(B)}{12} = \frac{\sin(30^\circ)}{6\sqrt{2}}

. Solving for

\sin(B)

yields

\sin(B) = \frac{\sqrt{2}}{2}

. Because the question specifies that the angle is acute, the measure of the angle is

45^\circ

Step-by-Step Solution

Set up the Law of Sines relationship for triangle

ABC

using the side opposite angle

B

(

AC

) and the side opposite angle

A

(

BC

\frac{\sin(B)}{AC} = \frac{\sin(A)}{BC}

The Law of Sines states that the ratio of the sine of an angle to the length of its opposite side is constant for all three angles in a triangle.

Substitute the given values into the equation:

AC = 12

BC = 6\sqrt{2}

, and

A = 30^\circ

\frac{\sin(B)}{12} = \frac{\sin(30^\circ)}{6\sqrt{2}}

This allows us to solve for the single unknown variable, the sine of angle

B

Simplify the expression using the trigonometric value

\sin(30^\circ) = 0.5

and isolate

\sin(B)

\sin(B) = \frac{12 \cdot 0.5}{6\sqrt{2}} = \frac{6}{6\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

Simplifying the fractions helps us identify the standard trigonometric value.

Find the acute angle

B

whose sine value is

\frac{\sqrt{2}}{2}

B = 45^\circ

The inverse sine of

\frac{\sqrt{2}}{2}

for an acute angle is

45^\circ

Key Concept

Law of Sines

Estimated Time:1m 0s

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