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If the triangle ABC is inscribed in semi-circle BAC as above

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by Max@Math Revolution » Sun Mar 06, 2016 4:01 pm
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If the triangle ABC is inscribed in semi-circle BAC as above figure and BC is a diameter, the length of AB is 6 and the length of AC is 8, what is the length of arc BAC?

A. 5Ï€(pi)
B. 6Ï€(pi)
C. 7Ï€(pi)
D. 8Ï€(pi)
E. 10 π(pi)


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If the triangle ABC is inscribed in semi-circle BAC as above

by Brent@GMATPrepNow » Mon Mar 07, 2016 11:22 am
Max@Math Revolution wrote:Image

If the triangle ABC is inscribed in semi-circle BAC as above figure and BC is a diameter, the length of AB is 6 and the length of AC is 8, what is the length of arc BAC?

A. 5π
B. 6π
C. 7π
D. 8π
E. 10π
Since BC is the diameter of the semi-circle, we know that ∠BAC is 90º
For more on this circle property, watch this free video - https://www.gmatprepnow.com/module/gmat ... /video/880

In other words, we can conclude that BAC is a RIGHT TRIANGLE and side BC is the HYPOTENUSE.
This means we can apply the Pythagorean Theorem to get: 6² + 8² = (side BC)²
Simplify: 36 + 64 = (side BC)²
Simplify: 100 = (side BC)²
So, side BC = 10
In other words, the DIAMETER = 10

Circumference of COMPLETE circle = (DIAMETER)(π)
So, circumference of SEMIcircle = (DIAMETER)(π)/2
= (10)(π)/2
= 5π

Answer: A

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Mon Apr 30, 2018 8:59 am, edited 2 times in total.
Brent Hanneson - Creator of GMATPrepNow.com
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by Max@Math Revolution » Tue Mar 08, 2016 4:03 pm
If the triangle ABC is inscribed in semi-circle BAC as above figure and BC is a diameter, the length of AB is 6 and the length of AC is 8, what is the length of arc BAC?

A. 5Ï€(pi)
B. 6Ï€(pi)
C. 7Ï€(pi)
D. 8Ï€(pi)
E. 10 π(pi)


->Since BC is a diameter, angle A is 90 degrees. According to Pythagoras' theorem, AB^2+AC^2=BC^2 -> 6^2+8^2=10^2 and BC=10. Then, a circumference with a diameter 10 is 10Ï€(pi). The question asks arc BAC, which is 10Ï€/2=5Ï€. Thus, A is the answer.
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by rdds » Thu Mar 10, 2016 12:27 am
Angle A = 90 degree also it is a 3,4,5 triangle
so diameter is 10, r = 10/2 = 5
Circumference of a circle = 2 pi * r
semi-circle = pi*r= 5*pi
Answer : A
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