Two complementary angles, A and B, have measures in the ratio of 7 to 23, respectively. What is the ratio of the measure of the complement of angle A to the measure of the complement of angle B? Express your answer as a common fraction.
B is the complement of A, so
B/A = 69/21 =23/7
Surprise, surprise!
Let's start by finding the measures of angles A and B. Since the ratio of their measures is 7:23, we can set up the equation:
7x + 23x = 90
Where x is a common multiplier. Simplifying the equation:
30x = 90
Dividing both sides by 30:
x = 3
Now we can find the measures of angles A and B:
Angle A: 7x = 7 * 3 = 21
Angle B: 23x = 23 * 3 = 69
The measures of the complements of angles A and B can be found by subtracting their corresponding angles from 90:
Complement of angle A: 90 - 21 = 69
Complement of angle B: 90 - 69 = 21
Now, let's find the ratio of the measure of the complement of angle A to the measure of the complement of angle B:
Ratio = (Complement of Angle A) / (Complement of Angle B)
Ratio = 69/21
Simplifying the fraction:
Ratio = 23/7
Therefore, the ratio of the measure of the complement of angle A to the measure of the complement of angle B is 23/7.
To solve this problem, we first need to determine the measures of angles A and B.
Let's assume that angle A measures x degrees. Since angle A and B are complementary, we know that angle A + angle B = 90 degrees. Therefore, angle B must measure (90 - A) degrees.
According to the given information, the ratio of the measures of angles A and B is 7 to 23. So, we can set up the equation:
x degrees / ((90 - x) degrees) = 7/23
To solve for x, we can cross-multiply:
23x = 7(90 - x)
23x = 630 - 7x
30x = 630
x = 21
So, angle A measures 21 degrees, and angle B measures (90 - 21) = 69 degrees.
Now, we need to find the measures of the complements of angles A and B.
The complement of angle A is (90 - A) degrees = (90 - 21) = 69 degrees.
The complement of angle B is (90 - B) degrees = (90 - 69) = 21 degrees.
Finally, we can find the ratio of the measure of the complement of angle A to the measure of the complement of angle B:
(69 degrees) / (21 degrees) = 69/21 = 23/7
Therefore, the ratio of the measure of the complement of angle A to the measure of the complement of angle B is 23/7.
A =7/30 *90 degrees
A = 21 degrees
B= 23/30 *90
= 69 degrees