Question

Two circles touch each other externally at C and a common tangent touches them at A and B. Which one is true?

• A. $CD\perp AB$
• B. $\angle ACB={90}^o$
• C. $AC=BC$
• D. All of these

Answer 1

Answer: (B)

$\angle ACB={90}^o$

According to the question

Lets suppose that,

X and Y are two circle touch each other at P.

AB is the common tangent to circle X and Y at point A and B.

According In the given figer,

In triangle $PAC,\angle CAP=\angle APC=\alpha$

Similarly $CB=CP,\angle CPB=\angle PBC=\beta$

Now triangle APB,

$\angle PAB+\angle PBA+\angle APB=180$

$\alpha +\beta +(\alpha +\beta )=180$

$2\alpha +2\beta =180$

$\alpha +\beta =90$

$\therefore \angle APB=90=\alpha +\beta .$

This is the required solution.