Diameter Calculation for Semicircle Geometry Problem
Given a semicircle with diameter AB, point C on AB (AC = 6 cm), and point D on the semicircle such that CD is perpendicular to AB (CD = 9 cm). We need to find the diameter AB.
Altitude Theorem for Semicircles
The altitude CD from point D on the semicircle to the diameter AB divides the diameter into segments AC and CB. The altitude theorem states that the square of the altitude is equal to the product of the lengths of the segments of the diameter:
$ CD^2 = AC \times CB $
Calculation Steps
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Identify the known values from the problem statement:
- Length of segment AC = 6 cm
- Length of altitude CD = 9 cm
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Substitute the known values into the altitude theorem formula:
$ 9^2 = 6 \times CB $
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Calculate the length of the segment CB:
$ 81 = 6 \times CB $
$ CB = \frac{81}{6} $
$ CB = \frac{27}{2} = 13.5 \text{ cm} $
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Calculate the total length of the diameter AB by adding the lengths of the segments AC and CB:
$ AB = AC + CB $
$ AB = 6 \text{ cm} + 13.5 \text{ cm} $
$ AB = 19.5 \text{ cm} $
Result
The diameter of the semicircle is 19.5 cm.
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