Geometry quadrilateral Abcd inscribed in circle

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Quadrilateral ABCD is inscribed in circle O. AB=12, BC=16, CD=10 and <ABC is a right angle. Find the measure of like Ad in simplified radical form.

May 12th, 2015

First, notice that <ADC must also be 90 degrees because if the quadrilateral is inscribed in a circle then opposite angles must sum to 180; thus <abc + <adc = 180 --> <ADC = 90. Now we can see that triangles ABC and ACD share a common hypotenuse and so if we find the hypotenuse, AC, then we can use the Pythagorean theorem to find AD, because the we will know 2 sides of the right triangle ADC. So, now we can notice that triangle ABC is similar to a 3,4,5 triangle (its sides are greater by a factor of 4). Thus, AC = 20. Alternatively, if you do not see this you can also do it using the Pythagorean theorem although it is slower. Now we know that AD^2 + DC^2 = AC^2, by the Pythagorean theorem. Plugging in numbers yields: AD^2 + 10^2 = 20^2. Solving for AD gives: AD^2 = 300. AD = sqrt(300) = sqrt(10^2*3) = 10*sqrt(3). 

May 12th, 2015

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May 12th, 2015

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