Congruent Tangents and Circumscribed Polygons

Study Guide

Key Definition

A $\text{tangent}$ to a circle is a line that touches the edge of the circle without actually cutting through it. If two tangents emanate from the same exterior point, they are congruent. A circumscribed polygon is one that encloses a circle, with each side of the polygon tangent to the circle.

Important Notes

Tangents from the same point are always congruent.
A polygon is circumscribed if it encloses a circle perfectly, with each side of the polygon tangent to the circle.
The term 'congruent' in geometry essentially means 'equal'.
In geometry, a secant is a line that cuts through a circle.

Mathematical Notation

$AB \cong AC$ means line AB is congruent to line AC.

Remember to use proper notation when solving problems

Why It Works

The property that tangents from the same point are congruent is based on the properties of circles and right triangles. When a tangent is drawn to a circle, it forms a right angle with the radius at the point of tangency. This creates two right triangles when two tangents are drawn from the same point, which are congruent by the Hypotenuse-Leg theorem.

Remember

Remember that a polygon is 'circumscribed' around a circle if each side of the polygon is tangent to the circle.

Quick Reference

Congruent Tangents:$AB \cong AC$

Understanding Congruent Tangents and Circumscribed Polygons

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Beginner

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Beginner Explanation

A tangent to a circle is a line that touches the circle at one point. If two tangents are drawn from the same point outside the circle, they are congruent.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

If lines $AB$ and $AC$ are both tangent to circle $O$ from point $A$, which statement is true?

Real-World Problem

Question Exercise

Intermediate

Designing a Garden

A gardener wants to design a flower bed in the shape of a pentagon, with a circular walkway touching each side of the pentagon. Can this be considered a circumscribed polygon?

Recap

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