Angle A has a measure of d degrees, and angle B has a measure of r radians. The sum of the… — SAT

Angle $A$ has a measure of $d$ degrees, and angle $B$ has a measure of $r$ radians. The sum of the degree measure of angle $A$ and the degree equivalent of the measure of angle $B$ is $180$ . If the measure of angle $A$ , in degrees, is $3$ times the degree equivalent of the measure of angle $B$ , and $r = k\pi$ , what is the value of $k$ ?

Answer: 0.25

Answer

The value of

k

0.25

(or

\frac{1}{4}

The correct answer is

0.25

(or

\frac{1}{4}

). To find this, we solve the system of equations representing the degree relationship:

d + D_B = 180

and

d = 3D_B

, where

D_B

is the degree equivalent of angle

B

. This gives

D_B = 45

. Converting

45^\circ

to radians by multiplying by

\frac{\pi}{180}

gives

\frac{\pi}{4}

radians, which means the coefficient

k

0.25

Step-by-Step Solution

Set up a system of equations using the given information.

d + D_B = 180

and

d = 3D_B

, where

d

is the degree measure of angle

A

and

D_B

is the degree equivalent of angle

B

To translate the verbal descriptions of the relationships between the angle measures into mathematical equations.

Solve the system of equations for

D_B

D_B = 45

Substituting

d = 3D_B

into the first equation yields

3D_B + D_B = 180

, which simplifies to

4D_B = 180

. Dividing both sides by

4

gives

D_B = 45

Convert the degree measure of angle

B

to radians.

r = \frac{\pi}{4}

radians

To convert degrees to radians, multiply the degree measure by

\frac{\pi}{180}

. This gives

45 \times \frac{\pi}{180} = \frac{\pi}{4}

Find the value of

k

from the expression

r = k\pi

k = 0.25

(or

\frac{1}{4}

)

Since

r = \frac{\pi}{4} = 0.25\pi

, comparing this to

r = k\pi

shows that

k = 0.25

Key Concept

Converting degrees to radians

Question

Answer

Step-by-Step Solution

Key Concept

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