If a circle circumscribes a triangle bounded by the x-axis, y-axis and

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

Since the triangle is bounded by both axes origin (0,0) is one of the vertices of the triangle.
At x-axis, y = 0,
For 3y - 4x = -24, x = 6 So another of the vertices of the triangle is point (6,0)

At y-axis, x = 0
For 3y - 4x = -24, y = -8 So third of vertices of the triangle is point (0,-8)

So, the triangle is a right angled triangle at origin (0,0) with side lengths 6(x-axis) and 8(y-axis).

As circle is circumscribing the triangle, the hypotenuse is the diameter of the triangle.
Hypotenuse = \(\sqrt{6^2 + 8^2}\) = 10

Radius of circle = \(\frac{Hypotenuse of triangle }{ 2} = \frac{10}{2}\) = 5

Answer D.

The intercepts of the line 3y - 4x = -24 are (0,-8), (6,0). Hence the lengths of the sides of the triangle are 6,8, and 10.
Radius of a circle that circumscribes the triangle of side 6,8,10 = 6*8*10/(4A) where A is the area of the triangle = 6*8/2 = 24
Radius = 480/96 = 5.

The answer is D.

Note: The hypotenuse is the diameter of the circle [Angle inside a semicircle is 90 degrees]

For the line 3y - 4x = -24,
To find x-intercept: Put y = 0;
--> -4x = -24
--> x = 6

To find y-intercept: Put x = 0;
--> 3y = -24
--> y = -8

So, the opposite ends of the diameter are (6, 0) & (0, -8)
--> Diameter = \(\sqrt{(6 - 0)^2 + (0 - (-8))^2} = \sqrt{100} = 10\)
--> Radius of the circle = 10/2 = 5

Option D

The circle circumscribes the right triangles -with sides of 6, 8 and 10 (hypotenuse)- at (0,0), (6,0), and (0,-8).

SINCE this is a right triangle -90 degrees at (0,0)-, the hypotenuse of the triangle MUST BE equal to the diameter of the circle, which is 10.

Thus, the radius of the circle is 5.

FINAL ANSWER IS (D)

Posted from my mobile device

If a circle circumscribes a right angle triangle then the longest side (hypotenuse) is the diameter of the circle.
Here the sides are 6,8 and 10 (considering the positive distances)
X intercept is 6
Y intercept is -8
So diameter=10, radius = 5
D is the answer

Posted from my mobile device

3y - 4x = -24
Touches X axis at (6,0)
Touches y axis at (0,-8)

As it's a right angle triangle, circumcentre lies at the midpoint of the hypotenuse.....

Hypotenuse of triangle= Diameter of circumcircle= √8^2+6^2= 10

Radius=5

OA:D

3y-4x=-24, 3y=4x-24, y=4/3(x)-8
y=0: 4x/3=8, x=3*8/4=6
x=0: y=-8
right triangle sides 8 and 6
hypotenuse: 8^2+6^2=64+36=100^(1/2)=10
rule: for any right triangle, the hypotenuse is a diameter of the circumcircle
d=10, r=5

Ans (D)

3y - 4x = 24 --> 3y=4x-24 --> y=4x/3-8
intersection point with y-axis (x=0) --> y=8
intersection point with x-axis (y=0) --> x=6
The triangle is a right triangle with sides 6 and 8 --> the hypotenuse of the right triangle is the diamater of the circle
Diameter= (8^2+6^2)^(1/2)=10
Radius=Diameter/2=5 --> Hence D

The circle will be formed with intercept on x axis as 6,0 and y axis as 0,-8
since x and y axis are perpendicular to each other
the hypotenuse will be
\(\sqrt{ 6^2 +8^2}\)
= 10
Also since the axis are perpendicular to each other hypotenuse is the diameter of a circle = 10
hence radius
= 5 cm = 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 ?

if x= 0, then y = —8 (0,—8)
if y= 0, then x = 6 (6, 0)
—> the triangle should be right-angled triangle. Its hypotenuse is 10.
—> The hypotenuse of any right-angled triangle inscribed in the circle must be diameter of that circle.
—> R = 10/2 = 5

Answer (D)

Posted from my mobile device

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