In the figure shown, the triangle

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boomgoesthegmat: Senior | Next Rank: 100 Posts; Posts: 93; Joined: Mon Apr 25, 2016 2:22 pmFollowed by: 1 member; Thanked: 1 times

by boomgoesthegmat » Wed May 04, 2016 4:47 am

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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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Source: Beat The GMAT — Problem Solving | Wed May 04, 2016 4:47 am

DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 amFollowed by: 128 members; Location: Boston, MA; Thanked: 1153 times; GMAT Score:770

by DavidG@VeritasPrep » Wed May 04, 2016 5:06 am

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

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.

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DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 amFollowed by: 128 members; Location: Boston, MA; Thanked: 1153 times; GMAT Score:770

by DavidG@VeritasPrep » Wed May 04, 2016 5:11 am

See here for another good question that tests this principle: https://www.beatthegmat.com/inscribed-tr ... 74152.html

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Re: In the figure shown, the triangle

by [email protected] » Wed Apr 21, 2021 8:05 pm

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

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Rich

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