Three circles have centres $A, B$ and $C$ with radii 2, 4 and 6 respectively. The circles are tangent to each other as shown. Triangle $A B C$ has
(A) $\angle A$ obtuse
(B) $\angle B=90^{\circ}$
(C) $\angle A=90^{\circ}$
(D) all angles acute
(E) $\angle B=\angle C$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-139.jpg?height=371&width=288&top_left_y=2205&top_left_x=1363)
Three circles have centres $A, B$ and $C$ with radii 2,4 and 6 respectively. The circles are tangent to each other as shown. Triangle $A B C$ has
(A) $\angle A$ obtuse
(B) $\angle B=90^{\circ}$
(C) $\angle A=90^{\circ}$
(D) all angles acute
(E) $\angle B=\angle C$

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-280.jpg?height=428&width=320&top_left_y=691&top_left_x=1336)

## Solution

Since the circles are mutually tangent, the lines joining their centres pass through the points of tangency. Thus, the sides of $\triangle A B C$ have lengths 6 , 8 and 10 if we write in the radii as shown. Since $10^{2}=6^{2}+8^{2}$, the triangle is right-angled at $A$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-280.jpg?height=406&width=401&top_left_y=1190&top_left_x=1233)

ANSWER: (C)