The ratio of two complementary angles is 3:7. Find the measures of both angles.
Let the two complementary angles be denoted by \(\alpha\) and \(\beta\).
We are given that the angles are complementary, which means their sum is \(90^\circ\):
\(\alpha + \beta = 90^\circ\).
We are also given that the ratio of the two angles is \(3:7\).
This means we can express the angles in terms of a common variable \(x\):
\(\alpha = 3x\)
\(\beta = 7x\)
where \(x\) is a positive constant.
Substitute the expressions for \(\alpha\) and \(\beta\) into the sum equation:
\(3x + 7x = 90^\circ\)
Combine the like terms on the left side:
\(10x = 90^\circ\)
To find the value of \(x\), divide both sides of the equation by 10:
\(x = \frac{90^\circ}{10} = 9^\circ\)
Now that we have found the value of \(x = 9^\circ\), we can calculate the measures of the two angles:
\(\alpha = 3x = 3 \times 9^\circ = 27^\circ\)
\(\beta = 7x = 7 \times 9^\circ = 63^\circ\)
We can verify the sum:
\(27^\circ + 63^\circ = 90^\circ\)
And the ratio:
\(\frac{27}{63} = \frac{3 \times 9}{7 \times 9} = \frac{3}{7}\)
Both conditions are satisfied. The angles are \(27^\circ\) and \(63^\circ\).
In architecture and engineering, finding complementary angles is useful when designing support structures or calculating slopes. For example, if you are designing a ramp where the total angle of a corner must be \(90^\circ\), you might need to divide the space into two parts with a specific proportion for aesthetics or functionality. Knowing how to use ratios to determine these specific angles ensures that the structure meets precise design requirements.