Question:
In a triangle \(ABC\), the lengths of the sides \(AB\) and \(AC\) equal \(17.5\) cm and \(9\) cm respectively. Let \(D\) be a point on the line segment \(BC\) such that \(AD\) is perpendicular to \(BC\). If \(AD = 3\) cm, then what is the radius (in cm) of the circle circumscribing the triangle \(ABC\)?
In a triangle \(ABC\), the lengths of the sides \(AB\) and \(AC\) equal \(17.5\) cm and \(9\) cm respectively. Let \(D\) be a point on the line segment \(BC\) such that \(AD\) is perpendicular to \(BC\). If \(AD = 3\) cm, then what is the radius (in cm) of the circle circumscribing the triangle \(ABC\)?
Show Hint
In the right triangles \(ABD\) and \(ACD\), find the sines of the angles at \(B\) and \(C\), then use the law of sines \(a / \sin A = 2R\).
Updated On: Nov 4, 2026
- 17.05
- 27.85
- 26.25
- 32.25
- 22.45
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Find the sine of angle C:
In the right triangle \(ADC\) (right angle at \(D\)), \(\sin C = \dfrac{AD}{AC} = \dfrac{3}{9} = \dfrac{1}{3}\).
Step 2: Use the law of sines on side AB:
In triangle \(ABC\), \(\dfrac{AB}{\sin C} = 2R\), where \(R\) is the circumradius. So \(2R = \dfrac{17.5}{1/3} = 52.5\).
Step 3: Find R:
\(R = \dfrac{52.5}{2} = 26.25\) cm.
Step 4: Check using angle B:
In the right triangle \(ABD\), \(\sin B = \dfrac{3}{17.5} = \dfrac{6}{35}\). Then \(2R = \dfrac{AC}{\sin B} = \dfrac{9 \times 35}{6} = 52.5\), so \(R = 26.25\) cm again.
Final Answer:
The correct option is (3): \(26.25\).
In the right triangle \(ADC\) (right angle at \(D\)), \(\sin C = \dfrac{AD}{AC} = \dfrac{3}{9} = \dfrac{1}{3}\).
Step 2: Use the law of sines on side AB:
In triangle \(ABC\), \(\dfrac{AB}{\sin C} = 2R\), where \(R\) is the circumradius. So \(2R = \dfrac{17.5}{1/3} = 52.5\).
Step 3: Find R:
\(R = \dfrac{52.5}{2} = 26.25\) cm.
Step 4: Check using angle B:
In the right triangle \(ABD\), \(\sin B = \dfrac{3}{17.5} = \dfrac{6}{35}\). Then \(2R = \dfrac{AC}{\sin B} = \dfrac{9 \times 35}{6} = 52.5\), so \(R = 26.25\) cm again.
Final Answer:
The correct option is (3): \(26.25\).
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