A point is marked one quarter of the way along each side of a triangle, as shown.
What fraction of the area of the triangle is shaded?
Kindly provided by UKMT A point is marked one quarter of the way along each side of a triangle, as shown. What fraction of the area of the triangle is shaded?
11 Comments
Andrey Dyukmedzhiev
5/3/2016 11:00:57 am
Using the formula about the area of a triangle, expressed by 2 sides and angle in-between, we obtain the ratio of the areas of each small white triangle and the large one (1/4)*(3/4) = 3/16. Hence the area of the red-shaded quadrilateral is 1 - 2*(3/16) = 10/16 of the area of the enclosing triangle.
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Ahmad
5/3/2016 01:41:22 pm
yes, me too ..the same Mr. Andrey ...5/8
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Gurpreet Singh
6/3/2016 01:40:14 am
Yes it is 5/8
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Paul
6/3/2016 06:41:03 am
Have you each used the same method to arrive at your answer?
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Ahmad
7/3/2016 03:45:29 am
Yes, it is the same idea.
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Andrey Dyukmedzhiev
8/3/2016 01:11:37 pm
Thank you Paul! A more interesting well-known problem: connect each vertex with the black point on the opposite side. What is the ratio of the areas of the central triangle and initial one?
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tom
7/3/2016 02:23:29 am
Call the points of the original triangle ABC and the quarter points on AB, BC, and AC respectively X,Y,Z. The triangles AXZ, BYX and CZY are each 3/16 times the size of ABC, so XYZ has area 7/16 of ABC; so the quadrangle AXYZ has area XYZ+AXZ, and that's 10/16 of ABC.
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Ahmad
7/3/2016 06:59:49 am
Today, I read theorem which name "Ceva's Theorem", and some of it's properties allowing us to calculate area of XYZ (triangle which Mr. Tom mentioned above )but directly as a ratio of area ABC by formula, and it will be 7/16...and by adding 3/16 , it will be done...
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Simon Geard
7/3/2016 10:21:56 pm
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Simon Geard
7/3/2016 10:30:42 pm
Okay, so my attempt to have a nicely formatted answer has failed, it got truncated mid-paste. I have uploaded the file to
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