10 7 Special Segments in a Circle Objectives

10. 7 Special Segments in a Circle

Objectives Find measures of segments that intersect in the interior of a circle. Find measures of segments that intersect in the exterior of a circle.

Segments in a Circle Theorem 10. 15: If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal. A AO • OB = CO • OD C D O B

Example 1: Find x. Theorem 10. 15 Multiply. Divide each side by 8. Answer: 13. 5

Your Turn: Find x. Answer: 12. 5

Example 2: Biologists often examine organisms under microscopes. The circle represents the field of view under the microscope with a diameter of 2 mm. Determine the length of the organism if it is located 0. 25 mm from the bottom of the field of view. Round to the nearest hundredth.

Example 2: Draw a model using a circle. Let x represent the unknown measure of the equal lengths of the chord which is the length of the organism. Use the products of the lengths of the intersecting chords to find the length of the organism. Note that

Example 2: Segment products Substitution Simplify. Take the square root of each side. Answer: 0. 66 mm

Your Turn: Phil is installing a new window in an addition for a client’s home. The window is a rectangle with an arched top called an eyebrow. The diagram below shows the dimensions of the window. What is the radius of the circle containing the arc if the eyebrow portion of the window is not a semicircle? Answer: 10 ft

Segments Outside of a Circle Theorem 10. 16: If two secants intersect outside a circle, then the product of the measures of the external secant segment and the entire secant segment is equal to the product of the measures of the other external secant segment and its secant segment. Z OW • OZ = OY • OX W O Y X

Example 3: Find x if EF 10, EH 8, and FG 24.

Example 3: Secant Segment Products Substitution Distributive Property Subtract 64 from each side. Divide each side by 8. Answer: 34. 5

Your Turn: Find x if Answer: 26 and

Segments Outside of a Circle Theorem 10. 17: If a tangent segment and a secant segment intersect outside a circle, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external segment. Z OZ • OZ = OX • OY O Y X

Example 4: Find x. Assume that segments that appear to be tangent are tangent. Disregard the negative solution. Answer: 8

Your Turn: Find x. Assume that segments that appear to be tangent are tangent. Answer: 30

Assignment Pre-AP Geometry Pg. 572 #8 - 30 Geometry: Pg. 572 #8 – 19, 22 - 28