In the given figure two circles touch each other internally in a point A. The radius of the smaller circle with centre M is 5.
The smaller circle passes through the centre N of the larger circle. The tangent to the smaller circle drawn through C intersects the larger circle in point D. Find CD.

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Solution

CA is a secant and CP is a tangent to the smaller circle .
By secant-tangent property, we have:
$CN \times CA = {CP}^{2} \Rightarrow 10 \times 20 = {CP}^{2} \Rightarrow CP = 10 \sqrt{2} units$

$Now, we have : ∠ CPM = 90 ° (Radius is perpendicular to the tangent) ∠ ADC = 90 ° (Angle in a semicircle is a right angle) ∴ ∠ CPM = ∠ ADC Since they are corresponding angles and are equal, PM ∥ DA . In ∆ CPM and ∆ CDA, we have : ∠ CPM = ∠ CDA = 90 ° ∠ MCP = ∠ ACD (Common) ∴ ∆ CPM ~ ∆ CDA (BY AA similarity) i . e ., \frac{CP}{CD} = \frac{CM}{CA} \Rightarrow CD = \frac{CP \times CA}{CM} = \frac{10 \sqrt{2} \times 20}{15} = \frac{40}{3} \sqrt{2} units$

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In the given figure two circles touch each other internally in a point A. The radius of the smaller circle with centre M is 5. The smaller circle passes through the centre N of the larger circle. The tangent to the smaller circle drawn through C intersects the larger circle in point D. Find CD.

In the given figure two circles touch each other internally in a point A. The radius of the smaller circle with centre M is 5.
The smaller circle passes through the centre N of the larger circle. The tangent to the smaller circle drawn through C intersects the larger circle in point D. Find CD.