CBSE Class 10 Mathematics Standard 2023: question 33(A)

Hints

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Step-by-step solution

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1. Figure: Draw $△ABC$ with D a point on BC. Join AD. Mark $\angle BAC$ at vertex A and $\angle ADC$ at vertex D as equal angles.

   Given: $\angle ADC=\angle BAC$. To prove: $C{A}^2=CB\cdot CD$.

   1 mark

2. Consider $△CBA$ and $△CAD$.

   $\angle BCA=\angle ACD$, since $\angle C$ is common to both triangles.

   1 mark

3. $\angle BAC=\angle ADC$ [given]

   1 mark

4. So $△CBA\sim △CAD$ [AA similarity criterion].

   The correspondence is $C\leftrightarrow C$, $B\leftrightarrow A$, $A\leftrightarrow D$.

   1 mark

5. Corresponding sides of similar triangles are proportional:

   $\frac{CB}{CA}=\frac{CA}{CD}$

   $\Rightarrow C{A}^2=CB\cdot CD$. Hence proved.

   1 mark

Answer: $△CBA\sim △CAD$ (AA), so $\frac{CB}{CA}=\frac{CA}{CD}$, giving $C{A}^2=CB\cdot CD$.

Revise the chapter

• Chapter: Triangles (notes, formulas, where marks are lost and every past question)
• Topic: Problems on Similar Triangles