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If a circle circumscribes a triangle bounded by the \(x\)-axis, \(y\)-axis and the line \(3y - 4x = -24\), what is the radius of the circle ?
A. \(2.5\)
B. \(3\)
C. \(4\)
D. \(5\)
E. \(10\)
A. \(2.5\)
B. \(3\)
C. \(4\)
D. \(5\)
E. \(10\)
06 Dec 2021, 08:16
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Official Solution:
If a circle circumscribes a triangle bounded by the \(x\)-axis, \(y\)-axis and the line \(3y - 4x = -24\), what is the radius of the circle ?
A. \(2.5\)
B. \(3\)
C. \(4\)
D. \(5\)
E. \(10\)
The triangle we get will be a right triangle (\(x\) and \(y\) axis are at 90°). Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle's side, then that triangle is a right triangle. So, the hypotenuse of the triangle we get, will be be twice the radius of the circumscribing circle.
Let's find the hypotenuse.
The \(x\) and \(y\) intercepts of \(3y - 4x = -24\) are \((6, 0)\) and \((0, -8)\), respectively. So, the lengths of the legs of the triangle will be 6 and 8.
The hypotenuse, is thus \(\sqrt{6^2+8^2}=10\) and the radius is half of that, so 5.
Answer: D
If a circle circumscribes a triangle bounded by the \(x\)-axis, \(y\)-axis and the line \(3y - 4x = -24\), what is the radius of the circle ?
A. \(2.5\)
B. \(3\)
C. \(4\)
D. \(5\)
E. \(10\)
The triangle we get will be a right triangle (\(x\) and \(y\) axis are at 90°). Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle's side, then that triangle is a right triangle. So, the hypotenuse of the triangle we get, will be be twice the radius of the circumscribing circle.
Let's find the hypotenuse.
The \(x\) and \(y\) intercepts of \(3y - 4x = -24\) are \((6, 0)\) and \((0, -8)\), respectively. So, the lengths of the legs of the triangle will be 6 and 8.
The hypotenuse, is thus \(\sqrt{6^2+8^2}=10\) and the radius is half of that, so 5.
Answer: D
