Cyclic Quadrilaterals

New York State Common Core Math Geometry, Module 5, Lesson 20

Worksheets for Geometry

Student Outcomes

• Students show that a quadrilateral is cyclic if and only if its opposite angles are supplementary.
• Students derive and apply the area of cyclic quadrilateral ABCD as 1/2 AB·CD·sin(w) where w is the measure of the acute angle formed by diagonals AB and CD.

Cyclic Quadrilaterals

Classwork

Opening Exercise

Given cyclic quadrilateral 𝐴𝐵𝐶𝐷 shown in the diagram, prove that 𝑥 + 𝑦 = 180°

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Example

Given quadrilateral 𝐴𝐵𝐶𝐷 with 𝑚∠𝐴+ 𝑚∠𝐶 = 180°, prove that quadrilateral 𝐴𝐵𝐶𝐷 is cyclic; in other words, prove that points 𝐴, 𝐵, 𝐶, and 𝐷 lie on the same circle.

Exercises

1. Assume that vertex 𝐷′′ lies inside the circle as shown in the diagram. Use a similar argument to Example 1 to show that vertex 𝐷′′ cannot lie inside the circle.
2. Quadrilateral 𝑃𝑄𝑅𝑆 is a cyclic quadrilateral. Explain why △ 𝑃𝑄𝑇 ~ △ 𝑆𝑅𝑇.
3. A cyclic quadrilateral has perpendicular diagonals. What is the area of the quadrilateral in terms of 𝑎, 𝑏, 𝑐, and 𝑑 as shown?
4. Show that the triangle in the diagram has area 1/2 𝑎𝑏 sin(𝑤).
5. Show that the triangle with obtuse angle (180 − 𝑤)° has area 1/2 𝑎𝑏 sin(𝑤).
6. Show that the area of the cyclic quadrilateral shown in the diagram is Area = 1/2 (𝑎 + 𝑏)(𝑐 + 𝑑) sin(𝑤).

Lesson Summary

THEOREM:

Given a convex quadrilateral, the quadrilateral is cyclic if and only if one pair of opposite angles is supplementary. The area of a triangle with side lengths 𝑎 and 𝑏 and acute included angle with degree measure 𝑤: The area of a cyclic quadrilateral 𝐴𝐵𝐶𝐷 whose diagonals 𝐴𝐶 and 𝐵𝐷 intersect to form an acute or right angle with degree measure 𝑤:

Relevant Vocabulary

CYCLIC QUADRILATERAL: A quadrilateral inscribed in a circle is called a cyclic quadrilateral.

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