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

Question

Competition Mode Question

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

FRESH GMAT CLUB TESTS' QUESTION

Are You Up For the Challenge: 700 Level Questions

M37-100

Bunuel · Accepted Answer

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

GMATinsight · Answer

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

Video explanation is here. 
 https://www.youtube.com/watch?v=XCPf5PeGh80

unraveled · Answer

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.

eakabuah · Answer

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.

CareerGeek · Answer

Note: The hypotenuse is the diameter of the circle

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

freedom128 · Answer

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

Lenel · Answer

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

madgmat2019 · Answer

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

exc4libur · Answer

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)

cristianov · Answer

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

globaldesi · Answer

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

lacktutor · Answer

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

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards