Question Video: Determining Whether a Rhombus Is a Cyclic Quadrilateral | Nagwa Question Video: Determining Whether a Rhombus Is a Cyclic Quadrilateral | Nagwa

Question Video: Determining Whether a Rhombus Is a Cyclic Quadrilateral Mathematics • Third Year of Preparatory School

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True or False: All rhombuses are cyclic quadrilaterals.

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Video Transcript

True or False: All rhombuses are cyclic quadrilaterals.

We can begin by recalling that a cyclic quadrilateral is a quadrilateral that has all four vertices inscribed on a circle. A rhombus is defined as a quadrilateral with all four sides equal in length. In the figure below, we are helpfully given a rhombus. Notice that all four sides are congruent. We can use the angle properties to prove if a quadrilateral is cyclic or not. If we consider the angles at the diagonals, we can use the property that if an angle created by a diagonal and side is equal to the angle created by the other diagonal and opposite side, then the quadrilateral is cyclic.

So let’s consider some of the properties of a rhombus. There are two pairs of parallel sides. And as we can see on the diagram, the diagonals are perpendicular to each other. We can also say that they are perpendicular bisectors of each other. So let’s consider one of the angles in this rhombus. Let’s take the angle 𝐴𝐵𝐷. If this was a cyclic quadrilateral, then the measure of this angle 𝐴𝐵𝐷 would be equal to the measure of angle 𝐴𝐶𝐷. But we can’t actually prove that this is equal in measure. The only angle measure which is the same as angle 𝐴𝐵𝐷 is that of angle 𝐵𝐷𝐶. This comes from the fact that we have two parallel lines, 𝐴𝐵 and 𝐵𝐶, and the transversal 𝐵𝐷.

In the same way, we could prove that the angle measure of angle 𝐵𝐴𝐶 is equal to the angle measure of angle 𝐴𝐶𝐷. But this doesn’t help prove that the rhombus is cyclic. We would need either this pair of angles at the top or this pair of angles at the base to be equal in measure. In fact, the only way in which we could get these pairs of angles to be congruent would be when all four angles are congruent. In this case, the rhombus would have four interior angles of 90 degrees, and it would in fact be a square. So when a rhombus is a square, it’s a cyclic quadrilateral. But we can’t say that all rhombuses are cyclic. Therefore, we can give the answer false.

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