Rectangles Inscribed in Circles

New York State Common Core Math Geometry, Module 5, Lesson 3

Worksheets for Geometry

Student Outcomes

• Inscribe a rectangle in a circle.
• Understand the symmetries of inscribed rectangles across a diameter.

Rectangles Inscribed in Circles

Classwork

Opening Exercise

Using only a compass and straightedge, find the location of the center of the circle below. Follow the steps provided.

• Draw chord 𝐴𝐵.
• Construct a chord perpendicular to 𝐴𝐵 at endpoint 𝐵.
• Mark the point of intersection of the perpendicular chord and the circle as point 𝐶.
• 𝐴𝐶 is a diameter of the circle. Construct a second diameter in the same way.
• Where the two diameters meet is the center of the circle.

Explain why the steps of this construction work.

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Exploratory Challenge

Construct a rectangle such that all four vertices of the rectangle lie on the circle below.

Exercises

1. Construct a kite inscribed in the circle below, and explain the construction using symmetry.
2. Given a circle and a rectangle, what must be true about the rectangle for it to be possible to inscribe a congruent copy of it in the circle?
3. The figure below shows a rectangle inscribed in a circle.
   a. List the properties of a rectangle.
   b. List all the symmetries this diagram possesses.
   c. List the properties of a square.
   d. List all the symmetries of the diagram of a square inscribed in a circle.
4. A rectangle is inscribed into a circle. The rectangle is cut along one of its diagonals and reflected across that diagonal to form a kite. Draw the kite and its diagonals. Find all the angles in this new diagram, given that the acute angle formed by the diagonal of the rectangle in the original diagram was 40°.
5. Challenge: Show that the three vertices of a right triangle are equidistant from the midpoint of the hypotenuse by showing that the perpendicular bisectors of the legs pass through the midpoint of the hypotenuse.
   a. Draw the perpendicular bisectors of 𝐴𝐵 and 𝐴𝐶.
   b. Label the point where they meet 𝑃. What is point 𝑃?
   c. What can be said about the distance from 𝑃 to each vertex of the triangle? What is the relationship between the circle and the triangle?
   d. Repeat this process, this time sliding 𝐵 to another place on the circle and call it 𝐵′. What do you notice?
   e. Is there a relationship between 𝑚∠𝐴𝐵𝐶 and 𝑚∠𝐴𝐵′𝐶? Explain

Lesson Summary

Relevant Vocabulary

INSCRIBED POLYGON: A polygon is inscribed in a circle if all vertices of the polygon lie on the circle.

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