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Question:

Circle + Triangle

Given $AB$ is diameter, $C$ and $D$ lie on circumference, $AB = 15cm$, $AC = 12cm$, $BD = 9cm$, find area of quadrilateral ABCD.

Note that the points $O$ and $Q$ were not in the original diagram.

My attempt:

I guessed that $\triangle ABC$ is congruent to $\triangle ABD$, computed the area, and it actually came out to be the correct answer! But I have no idea how they can be proved congruent. And so, I need help in proving those triangles congruent.

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    $\begingroup$ Do you know that the angle in a semicircle is always a right angle? $\endgroup$ – Indrayudh Roy May 11 '14 at 18:39
  • $\begingroup$ @IndrayudhRoy indeed, I had solved such a question, a month ago, and i forgot it. Nevertheless, thanks for the hint :) $\endgroup$ – Gaurang Tandon May 11 '14 at 18:41
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    $\begingroup$ I hope you can solve it now. $\endgroup$ – Indrayudh Roy May 11 '14 at 18:42
  • $\begingroup$ Questions like these (Euclidean geometry type stuff) should be tagged as analytic-geometry. Algebraic geometry is completely different. $\endgroup$ – user98602 May 11 '14 at 18:52
  • $\begingroup$ @MikeMiller Thanks for the tip! Will keep in mind later on. $\endgroup$ – Gaurang Tandon May 11 '14 at 18:54
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Just notice that the angles $\angle ACB$ and $\angle ADB$ are right angles because they subtend the diameter of the circle. Since you know the measures of two of the sides of each of the two triangles, you may compute the measure of the third side using Pythagoras' theorem. Doing this you will show that they are indeed congruent by showing that the three sides have the same measures.

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  • $\begingroup$ Thanks! Acceptance is soon in 8 minutes ;) $\endgroup$ – Gaurang Tandon May 11 '14 at 18:42

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