Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L is
- 8
- 9
- 10
- 7
The Correct Option is B
Solution and Explanation
A triangle is given where the equation of one side is \(3x + 5y - 45 = 0\). We need to determine the radius of the circumcircle of this triangle, which is a right-angled triangle.
Concept Used:
In a right-angled triangle, the hypotenuse serves as the diameter of the circumcircle. Hence, the radius is:
Step 1: Find Intercepts
The equation \(3x + 5y = 45\) gives the intercepts:
- Put \(x = 0\): \(5y = 45 \Rightarrow y = 9\)
- Put \(y = 0\): \(3x = 45 \Rightarrow x = 15\)
So, the triangle has vertices at: \((0, 0), (0, 9), (15, 0)\)
Step 2: Compute Hypotenuse
The hypotenuse is between the points \((0, 9)\) and \((15, 0)\):
Step 3: Compute Radius
\[\text{Radius} = \frac{17.5}{2} \approx 8.75\]Rounding to the nearest integer: \(\boxed{9}\)
Final Answer:
The required radius (to the nearest integer) is \(\boxed{9}\).
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