Chapter 9: Q11. (page 336)

Given: $\bar{R S}$ is a common internal tangent to $⊙ A$ and $⊙ B$ .
Explain why $\frac{A C}{B C} = \frac{R C}{S C}$ .

Short Answer

Expert verified

$\frac{A C}{B C} = \frac{R C}{S C}$

Step by step solution

Step 1. Given information.

$\bar{R S}$ is a common internal tangent to the circles with centerA and B.

Step 2. Concept used.

If the line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangents.

This implies that,

$\begin{array}{l} A R ⊥ R S \\ S B ⊥ R S \end{array}$

Then,

$\begin{array}{l} ∠ A R C = 90 ° \\ ∠ B S C = 90 ° \end{array}$

Step 3. First prove that the two triangles are similar.

Consider in $Δ A R C$ and $Δ B S C$

$∠ A R C = ∠ B S C$ (Both the angles are of measure $90 °$ )

$∠ A C R = ∠ B C R$ (Vertical opposite angles is equal)

This implies that, $Δ A R C ~ Δ B S C$ (Angle-Angle Rule)

The corresponding sides in similar triangle are proportional.

Therefore, it is proved that, $\frac{A C}{B C} = \frac{R C}{S C}$

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Given: $\bar{R S}$ is a common internal tangent to $⊙ A$ and $⊙ B$ .
Explain why $\frac{A C}{B C} = \frac{R C}{S C}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept used.

Step 3. First prove that the two triangles are similar.

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Given: RS¯ is a common internal tangent to ⊙A and ⊙B.Explain why ACBC=RCSC.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept used.

Step 3. First prove that the two triangles are similar.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Pure Maths

Theoretical and Mathematical Physics

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Decision Maths

Study anywhere. Anytime. Across all devices.

Given: $\bar{R S}$ is a common internal tangent to $⊙ A$ and $⊙ B$ .
Explain why $\frac{A C}{B C} = \frac{R C}{S C}$ .