In quadrilateral ABCD, △A is a right angle, AB=35, BC=8, CD=6, and AD=5. Are points A, B, C, D concyclic?
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Step-by-Step Solution
Step 1: Draw the quadrilateral and diagonal
First, let's visualize the quadrilateral ABCD. We are given that ∠A is a right angle. This means that △ABD is a right-angled triangle. We can use the Pythagorean theorem to find the length of the diagonal BD.
Step 2: Calculate the length of diagonal BD
Substitute the given lengths of AB=35 and AD=5 into the Pythagorean theorem. Calculate BD2 and then take the square root to find the length of BD.
Step 3: Check if triangle BCD is a right-angled triangle
For points A, B, C, D to be concyclic, the quadrilateral must be cyclic. A property of cyclic quadrilaterals is that the sum of opposite angles is 180∘. Since ∠A=90∘, if the quadrilateral is cyclic, then ∠C must also be 90∘. Let's check if △BCD is a right-angled triangle using the Pythagorean theorem. If BC2+CD2=BD2, then ∠C=90∘.
Step 4: Compare the squares of the sides
We compare the sum of the squares of sides BC and CD with the square of the diagonal BD. We found BC2+CD2=100 and BD2=9250.
Step 5: Conclusion
Since BC2+CD2=BD2, △BCD is not a right-angled triangle, and therefore ∠C=90∘. Since the sum of opposite angles ∠A+∠C=180∘, the quadrilateral ABCD is not cyclic. Thus, points A, B, C, D are not concyclic.