# Properties of Tangents

### New York State Common Core Math Geometry, Module 5, Lesson 11

Worksheets for Geometry, Module 5, Lesson 11

Student Outcomes

• Students discover that a line is tangent to a circle at a given point if it is perpendicular to the radius drawn to that point.
• Students construct tangents to a circle through a given point.
• Students prove that tangent segments from the same point are equal in length.

Properties of Tangents

Classwork

Exercises

1. 𝐶𝐷 and 𝐶𝐸 are tangent to circle 𝐴 at points 𝐷 and 𝐸, respectively. Use a two-column proof to prove 𝑎 = 𝑏.
2. In circle 𝐴, the radius is 9 mm and 𝐵𝐶 = 12 mm.
a. Find 𝐴𝐶.
b. Find the area of △ 𝐴𝐶𝐷.
c. Find the perimeter of quadrilateral 𝐴𝐵𝐶𝐷.
3. In circle 𝐴, 𝐸𝐹 = 12 and 𝐴𝐸 = 13. 𝐴𝐸: 𝐴𝐶 = 1: 3.
a. Find the radius of the circle.
b. Find 𝐵𝐶 (round to the nearest whole number).
c. Find 𝐸𝐶

Lesson Summary

THEOREMS:

• A tangent line to a circle is perpendicular to the radius of the circle drawn to the point of tangency.
• A line through a point on a circle is tangent at the point if, and only if, it is perpendicular to the radius drawn to the point of tangency.

Relevant Vocabulary

• INTERIOR OF A CIRCLE: The interior of a circle with center 𝑂 and radius 𝑟 is the set of all points in the plane whose distance from the point 𝑂 is less than 𝑟.
A point in the interior of a circle is said to be inside the circle. A disk is the union of the circle with its interior.
• EXTERIOR OF A CIRCLE: The exterior of a circle with center 𝑂 and radius 𝑟 is the set of all points in the plane whose distance from the point 𝑂 is greater than 𝑟.
A point exterior to a circle is said to be outside the circle.
• TANGENT TO A CIRCLE: A tangent line to a circle is a line in the same plane that intersects the circle in one and only one point. This point is called the point of tangency.
• TANGENT SEGMENT/RAY: A segment is a tangent segment to a circle if the line that contains it is tangent to the circle and one of the end points of the segment is a point of tangency. A ray is called a tangent ray to a circle if the line that contains it is tangent to the circle and the vertex of the ray is the point of tangency.
• SECANT TO A CIRCLE: A secant line to a circle is a line that intersects a circle in exactly two points.
• POLYGON INSCRIBED IN A CIRCLE: A polygon is inscribed in a circle if all of the vertices of the polygon lie on the circle.
• CIRCLE INSCRIBED IN A POLYGON: A circle is inscribed in a polygon if each side of the polygon is tangent to the circle.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 