Geometry Question by Mithil Shah # 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 area of the triangle. Find the value of a+b if the answer is .
The answer is 199719.
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1 solution
Each section plus everything above that section base is a triangle. They are all similar triangles with the side lengths in the proportion. The length of base and height are also in the same proportion, of course. So, the areas are in the proportion.
The area of the triangle made up of the top two sections is {{{4}}} times the area of the smallest (top) triangle.
The area of the triangle made up of the top three sections is {{{9}}} times the area of the smallest (top) triangle.
Similarly,
The area of the triangle made up of the top nine sections is {{{81}}} times the area of the smallest (top) triangle.
The original triangle's area is {{{100}}} times the area of the smallest (top) triangle. Let's call the area of the smallest (top) triangle {{{x}}} square centimeters.
The area of the triangle made up of the top five sections is {{{81x}} square centimeters. The original triangle's area is {{{100x}}} square centimeters.
The difference, {{{100x-81x=19x}}} , is the area, in square centimeters of the bottom section. Our equation is {{{19x=1997}}} --> {{{x=1997/19}}}
The area of the original triangle is {{{100x=199700/19}}} square centimeters.