In the figure shown, the triangle is inscribed in the semicircle. If the length of line segment AB is 8 and the length of line segment BC is 6, what is the length of arc ABC?
A) 15pi
B) 12pi
C) 10pi
D) 7pi
E) 5pi
OA: E
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In the figure shown, the triangle
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boomgoesthegmat
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In the figure shown, the triangle is inscribed in the semicircle. If the length of line segment AB is 8 and the length of line segment BC is 6, what is the length of arc ABC?
A) 15pi
B) 12pi
C) 10pi
D) 7pi
E) 5pi
OA: E
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DavidG@VeritasPrep
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An inscribed angle opposite a circle's diameter is 90 degrees, so we know that angle ABC is a right angle. Moreover, we can see that triangle ABC is a 3:4:5 triangle, which means that if segment BC = 6 and AB = 8, then segment AC = 10. If the diameter is 10, the circumference of the full circle is 10pi. Here, arc ABC constitutes half the circle, or 5pi.boomgoesthegmat wrote:
In the figure shown, the triangle is inscribed in the semicircle. If the length of line segment AB is 8 and the length of line segment BC is 6, what is the length of arc ABC?
A) 15pi
B) 12pi
C) 10pi
D) 7pi
E) 5pi
OA: E
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DavidG@VeritasPrep
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- Posts: 2663
- Joined: Wed Jan 14, 2015 8:25 am
- Location: Boston, MA
- Thanked: 1153 times
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- GMAT Score:770
See here for another good question that tests this principle: https://www.beatthegmat.com/inscribed-tr ... 74152.html
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Hi All,
In this prompt, we are told that a triangle is INSCRIBED in a semi-circle (which means that all 3 points of the triangle are ON the circumference of the half-circle) – and we are given the lengths of the two shorter sides of the triangle (6 and 8). We’re asked for the ARC LENGTH of the semi-circle.
To answer this question, we’ll clearly need the radius of the semi-circle. There is also one specific math rule that we’ll need to know: when a triangle is inscribed in a circle – AND one of the sides is the DIAMETER of the circle – then we have a RIGHT TRIANGLE.
Since triangle ABC is a right triangle and the two ‘legs’ are 6 and 8, we have a 3/4/5 right triangle that has been ‘doubled’ (meaning that the sides are 6, 8 and 10). By extension, the diameter of the semi-circle is 10 and the radius is 5.
Circumference of a Circle is based on the formula:
C = (2)(pi)(Radius)
A semi-circle is half of a circle, so once we have the full circumference, we can divide that by 2 to find the arc length here:
(2)(pi)(5) / 2 = 5pi
Final Answer: E
GMAT Assassins aren’t born, they’re made,
Rich
In this prompt, we are told that a triangle is INSCRIBED in a semi-circle (which means that all 3 points of the triangle are ON the circumference of the half-circle) – and we are given the lengths of the two shorter sides of the triangle (6 and 8). We’re asked for the ARC LENGTH of the semi-circle.
To answer this question, we’ll clearly need the radius of the semi-circle. There is also one specific math rule that we’ll need to know: when a triangle is inscribed in a circle – AND one of the sides is the DIAMETER of the circle – then we have a RIGHT TRIANGLE.
Since triangle ABC is a right triangle and the two ‘legs’ are 6 and 8, we have a 3/4/5 right triangle that has been ‘doubled’ (meaning that the sides are 6, 8 and 10). By extension, the diameter of the semi-circle is 10 and the radius is 5.
Circumference of a Circle is based on the formula:
C = (2)(pi)(Radius)
A semi-circle is half of a circle, so once we have the full circumference, we can divide that by 2 to find the arc length here:
(2)(pi)(5) / 2 = 5pi
Final Answer: E
GMAT Assassins aren’t born, they’re made,
Rich
