Question
Download Solution PDFA line from point A is drawn that is tangent to the circle at point B. A secant is also drawn from point A to the circle intersecting it at points C and D. If AB = 42 cm and AC = 21 cm, then what is the ratio between AB and CD?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
A line from point A is tangent to the circle at point B.
A secant is drawn from point A to the circle intersecting it at points C and D.
AB = 42 cm (tangent segment)
AC = 21 cm (secant segment from A to C)
Formula Used:
The Power of a Point theorem states:
The square of the length of the tangent segment (AB) from a point to a circle is equal to the product of the lengths of the segments of the secant (AC and AD) from the same point.
AB2 = AC × AD
Calculation:
Let AD be the length of the secant segment from A to D.
According to the Power of a Point theorem:
AB2 = AC × AD
422 = 21 × AD
1764 = 21 × AD
AD = 1764 / 21
AD = 84 cm
CD = AD - AC
CD = 84 cm - 21 cm
CD = 63 cm
The ratio between AB and CD is:
AB : CD = 42 : 63
Simplify the ratio:
42 / 21 : 63 / 21
2 : 3
The ratio between AB and CD is 2 : 3.
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