Inscribed triangle

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pm; Thanked: 11 times; Followed by:8 members

Inscribed triangle

by gmattesttaker2 » Sat Aug 11, 2012 9:38 pm

Hello,

This is from P. 51 of MGMAT Geometry Guide. I was not clear with the explanation given:

"The right angle lies opposite a semi-circle, which is an arc that measures 180 degrees."

I was not sure how the arc is determined to be 180 degrees.

Can you please explain? Thanks a lot.

Best Regards,
Sri

P.S. I have attached the circle diagram.

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Source: Beat The GMAT — Problem Solving | Sat Aug 11, 2012 9:38 pm

adthedaddy: Master | Next Rank: 500 Posts; Posts: 167; Joined: Fri Mar 09, 2012 8:35 pm; Thanked: 39 times; Followed by:3 members

by adthedaddy » Sun Aug 12, 2012 2:42 am

This is based on a theorem "ANGLE IN A SEMI-CIRCLE IS A RIGHT ANGLE"
We learn that in High-school.

Following is the proof.

Refer attached diagram also.

1) We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.

2) We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.

3) But all of these angles together must add up to 180Â°, since they are the angles of the original big triangle.

Therefore x + y + x + y = 180, in other words 2(x + y) = 180.
and so x + y = 90. But x + y is the size of the angle we wanted to find.

Thus, for any angle inside a semi-circle, it is a right angle.
This a theorem and is universally true.

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