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.

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.

Substitute the given lengths of AB = (5)/(3) and AD = 5 into the Pythagorean theorem. Calculate BD² and then take the square root to find the length of BD.

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^°.

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^°.

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.

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.

In quadrilateral $ABCD$ , $\triangle A$ is a right angle, $AB=\dfrac{5}{3}$ , $BC=8$ , $CD=6$ , and $AD=5$ . Are points $A, B, C, D$ concyclic?

Answer: No, the points $A, B, C, D$ are not concyclic.

Step-by-step solution

Step 1: Draw the quadrilateral and diagonal

First, let's visualize the quadrilateral $ABCD$ . We are given that $\angle A$ is a right angle. This means that $\triangle ABD$ 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 $AB = \frac{5}{3}$ and $AD = 5$ into the Pythagorean theorem. Calculate $BD^2$ 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 $180^\circ$ . Since $\angle A = 90^\circ$ , if the quadrilateral is cyclic, then $\angle C$ must also be $90^\circ$ . Let's check if $\triangle BCD$ is a right-angled triangle using the Pythagorean theorem. If $BC^2 + CD^2 = BD^2$ , then $\angle C = 90^\circ$ .

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 $BC^2 + CD^2 = 100$ and $BD^2 = \frac{250}{9}$ .

Step 5: Conclusion

Since $BC^2 + CD^2 \neq BD^2$ , $\triangle BCD$ is not a right-angled triangle, and therefore $\angle C \neq 90^\circ$ . Since the sum of opposite angles $\angle A + \angle C \neq 180^\circ$ , the quadrilateral $ABCD$ is not cyclic. Thus, points $A, B, C, D$ are not concyclic.

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In quadrilateral ABCDABCDABCD, △A\triangle A△A is a right angle, AB=53AB=\dfrac{5}{3}AB=35​, BC=8BC=8BC=8, CD=6CD=6CD=6, and AD=5AD=5AD=5. Are points A,B,C,DA, B, C, DA,B,C,D concyclic?