Pantomiming

Geometry Level 2

The internal angles (measured in degrees) of a pentagon are in arithmetic progression. What is the value of the middle term?

Image Credit: Flickr Muslima2006 .


The answer is 108.

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Aquilino Madeira by Aquilino Madeira , 60, Portugal

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2 solutions

Ujjwala Ananth
Jun 22, 2015

Let middle term be a. Then Other Angles Are a+2d, a+d, a-d and a-2d. Sum of these angles is 5a. Sum of interior angles of polygon is 540° Therefore, 5a=540° I.e a=108°

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Moderator note:

Yes. Setting the middle term as a single variable is an underused technique when solving arithmetic progression related problems.

Chinmaya Chinmaya
Jun 28, 2015

Let the terms be a-2b, a-b, a, a+b, a+2b. Sum of internal angles of a polygon) =(n-2) 180° where n =no. Of sides. Sum of angles of a polygon =(5-2) 180°=3*180°=540°. So sum of angles = a-2b+a-b+a+a+b+a+2b =540°

> b, - b, 2b and - 2b cancel out.

So equation becomes: 5a=540° a=108° which is the middle term.

So 108° is the answer. :)

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