This is too easy.... Right?

Geometry Level 4

A circle touches two adjacent sides of a rectangle A B AB and A D AD at P P and Q Q respectively. The vertex C C of the rectangle lies on the circle. The length of the perpendicular from vertex C C to the chord P Q PQ is 5 2 \frac{5}{2} . Find the area of rectangle.

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The answer is 6.250.

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1 solution

Aniket Verma
Mar 4, 2015

One of the easy way to find the answer is taking the condition of the above figure which satisfies all the given conditions. Here the circle touches the rectangle at A A and O O , and side D D lies on the circle.

Here, D O = 5 2 DO = \frac{5}{2}

therefore A D = 5 2 2 AD = \frac{5\sqrt2}{2} , and C D = 5 2 2 CD = \frac{5}{2\sqrt2}

hence required area = = A D × C D AD\times CD = = ( 5 2 ) 2 (\frac{5}{2})^{2} = = 6.25 6.25 .

if anyone wants detailed solution than you are most welcome to ask me.

This is a special case where D is tangential and this extra condition is USED. So in general it may not be true. Though in this case it is true. What is the opinion of Challenge Master?

Niranjan Khanderia - 5 years, 2 months ago

Similar problem "Tricky geometry problem by Rayyan Shahid" has appeared. PQ was 5 there. In fact PQ/area is in invariant. That is Area=PQ 2 ^2 always. There are two good solutions there.

Niranjan Khanderia - 5 years, 2 months ago

i have a solution by coordinate geometry

Anubhav Tyagi - 4 years, 8 months ago

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