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

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If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post 31 Mar 2020, 01:15
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D
E

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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post Updated on: 31 Mar 2020, 02:11
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Quote:
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


Triangle bounded by X-axis, Y-Axis and Line 3y - 4x = -24

Has dimensions 8, 6, 10
i.e. a right triangle with hypotenuse = 10

CONCEPT: If a right triangle is inscribed in a circle then Hypotenuse of triangle = Diameter of Circle


i.e. Radius = 10/2 = 5

Answer: Option D

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Originally posted by GMATinsight on 31 Mar 2020, 01:51.
Last edited by GMATinsight on 31 Mar 2020, 02:11, edited 1 time in total.
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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post 31 Mar 2020, 02:00
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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

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.
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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post 31 Mar 2020, 02:00
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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.
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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post 31 Mar 2020, 02:05
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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
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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post 31 Mar 2020, 02:46
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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)

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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post 31 Mar 2020, 04:48
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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

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If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post Updated on: 01 Apr 2020, 06:39
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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

Originally posted by madgmat2019 on 31 Mar 2020, 08:10.
Last edited by madgmat2019 on 01 Apr 2020, 06:39, edited 1 time in total.
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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post 31 Mar 2020, 12:32
1
Quote:
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


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)
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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post 31 Mar 2020, 14:49
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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
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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post 31 Mar 2020, 20:45
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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
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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and  [#permalink]

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New post 31 Mar 2020, 21:26
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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 ?

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)

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Re: If a circle circumscribes a triangle bounded by the x-axis, y-axis and   [#permalink] 31 Mar 2020, 21:26

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

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