Triangle

Geometry Level 3

Nine lines drawn parallel to the base of a triangle divide the other two sides into 10 equal parts and the area into 10 distinct parts. If the area of the largest of these parts is 1997 sq.cms, find the area of the triangle. If ans is in the form of a b \frac ab , where a a and b b are coprime positive integers, submit a + b a+b as your answer.


The answer is 199719.

Yash Jain by Yash Jain , 21, India

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2 solutions

Marta Reece
Feb 22, 2017

The part of the triangle not included in the bottom trapezium is a triangle with a base 9 10 × b \frac{9}{10}\times b and height 9 10 × h \frac{9}{10}\times h (where b b and h h are the base and height of the original triangle) giving it area

9 10 × 9 10 A = 81 100 A \frac{9}{10}\times\frac{9}{10}A=\frac{81}{100}A .

What's left is the trapezium with area equal to 100 − 81 100 A = 19 100 A \frac{100-81}{100}A=\frac{19}{100}A therefore

1997 = 19 100 A 1997=\frac{19}{100}A and

A = 1997 × 100 19 A=\frac{1997\times 100}{19}

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Yash Jain
Jan 28, 2016

Let b,h be the base and height of the triangle resp. The largest part is a trapezium with sides b and 9b/10and height h/10. 》Area of the largest part is 1/2(h+9b/10)h/10 =1997 Therefore, Area of triangle =1/2bh=199700/19.

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