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More Lessons for Geometry

Common Core For Geometry

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**

- πΆπ· and πΆπΈ are tangent to circle π΄ at points π· and πΈ, respectively. Use a two-column proof to prove π = π.
- In circle π΄, the radius is 9 mm and π΅πΆ = 12 mm.

a. Find π΄πΆ.

b. Find the area of β³ π΄πΆπ·.

c. Find the perimeter of quadrilateral π΄π΅πΆπ·. - In circle π΄, πΈπΉ = 12 and π΄πΈ = 13. π΄πΈ: π΄πΆ = 1: 3.

a. Find the radius of the circle.

b. Find π΅πΆ (round to the nearest whole number).

c. Find πΈπΆ

**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 π and radius π is the set of all points in the plane
whose distance from the point π is less than π.

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 π and radius π is the set of all points in the plane
whose distance from the point π is greater than π.

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.

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