Chapter 4.3, Problem 28E

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7th Edition

Alexander + 2 others

ISBN: 9781337614085

Textbook Problem

Which type(s) of quadrilateral(s) is (are) necessarily cyclic?

a) A kite | b) A rectangle |

To determine

**To find:**

The type(s) of quadrilateral(s) is (are) necessarily cyclic.

Explanation

**Cyclic quadrilateral definition:**

Quadrilateral means four-sided figure. Put them together, and we get the definition for cyclic quadrilateral: any four-sided figure (quadrilateral) whose four vertices (corners) lie on a circle.

**Property of cyclic quadrilateral:**

The sum of opposite angles of a cyclic quadrilateral 180 degrees.

In other words, angle *A* + angle *C* = 180, and angle *B* + angle *D* = 180

**Calculation:**

To find a kite is necessarily cyclic or not:

When a kite is a cyclic quadrilateral (all four vertices lie on a circle), it must be composed of two congruent right triangles. This type of kite is known as a right kite.

A kite does have an in-circle i.e. a circle inscribed in it.

The cyclic kite and its in-circle is shown in given figure:

So, a kite is cyclic quadrilateral when its right kite.

To find a rectangle is necessarily cyclic or not:

If the quadrilateral is a cyclic quadrilateral, then all vertices of a quadrilateral lie on a circle...

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