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Tangent Lines

Lines perpendicular to the radius drawn to the point of tangency.

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Tangent Lines

Tangent Line Theorems

There are two important theorems about tangent lines.

1. Tangent to a Circle Theorem: A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.

\begin{align*}\overleftrightarrow{BC}\end{align*} is tangent at point \begin{align*}B\end{align*} if and only if \begin{align*}\overleftrightarrow{BC} \perp \overline{AB}\end{align*}.

This theorem uses the words “if and only if,” making it a biconditional statement, which means the converse of this theorem is also true.

2. Two Tangents Theorem: If two tangent segments are drawn to one circle from the same external point, then they are congruent.

\begin{align*}\overline{BC}\end{align*} and \begin{align*}\overline{DC}\end{align*} have \begin{align*}C\end{align*} as an endpoint and are tangent; \begin{align*}\overline{BC} \cong \overline{DC}\end{align*}.

What if a line were drawn outside a circle that appeared to touch the circle at only one point? How could you determine if that line were actually a tangent?

Examples

Example 1

Determine if the triangle below is a right triangle.

Use the Pythagorean Theorem. \begin{align*}4\sqrt{10}\end{align*} is the longest side, so it will be \begin{align*}c\end{align*}.

Does \begin{align*}8^2+10^2 \ & = \ \left ( 4\sqrt{10} \right )^2?\\ 64+100 & \ne 160\end{align*}

\begin{align*}\triangle ABC\end{align*} is not a right triangle. From this, we also find that \begin{align*}\overline{CB}\end{align*} is not tangent to \begin{align*}\bigodot A\end{align*}.

Example 2

If \begin{align*}D\end{align*} and \begin{align*}C\end{align*} are the centers and \begin{align*}AE\end{align*} is tangent to both circles, find \begin{align*}DC\end{align*}.

\begin{align*}\overline{AE} \perp \overline{DE}\end{align*} and \begin{align*}\overline{AE} \perp \overline{AC}\end{align*} and \begin{align*}\triangle ABC \sim \triangle DBE\end{align*} by AA Similarity.

To find \begin{align*}DB\end{align*}, use the Pythagorean Theorem.

\begin{align*}10^2+24^2&=DB^2\\ 100+576&=676\\ DB&=\sqrt{676}=26\end{align*}

To find \begin{align*}BC\end{align*}, use similar triangles. \begin{align*}\frac{5}{10}=\frac{BC}{26} \longrightarrow BC=13. \ DC=DB+BC=26+13=39\end{align*}

Example 3

\begin{align*}\overline{CB}\end{align*} is tangent to \begin{align*}\bigodot A\end{align*} at point \begin{align*}B\end{align*}. Find \begin{align*}AC\end{align*}. Reduce any radicals.

\begin{align*}\overline{CB}\end{align*} is tangent, so \begin{align*}\overline{AB} \perp \overline{CB}\end{align*} and \begin{align*}\triangle ABC\end{align*} a right triangle. Use the Pythagorean Theorem to find \begin{align*}AC\end{align*}.

\begin{align*}5^2+8^2&=AC^2\\ 25+64&=AC^2\\ 89&=AC^2\\ AC&=\sqrt{89}\end{align*}

Example 4

Using the answer from Example A above, find \begin{align*}DC\end{align*} in \begin{align*}\bigodot A\end{align*}. Round your answer to the nearest hundredth.

\begin{align*}DC & = AC - AD\\ DC & = \sqrt{89}-5 \approx 4.43\end{align*}

Example 5

Find the perimeter of \begin{align*}\triangle ABC\end{align*}.

\begin{align*}AE = AD, \ EB = BF,\end{align*} and \begin{align*}CF = CD\end{align*}. Therefore, the perimeter of \begin{align*}\triangle ABC=6+6+4+4+7+7=34\end{align*}.

\begin{align*}\bigodot G\end{align*} is inscribed in \begin{align*}\triangle ABC\end{align*}. A circle is inscribed in a polygon if every side of the polygon is tangent to the circle.

Review

Determine whether the given segment is tangent to \begin{align*}\bigodot K\end{align*}.

Find the value of the indicated length(s) in \begin{align*}\bigodot C\end{align*}. \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are points of tangency. Simplify all radicals.

\begin{align*}A\end{align*} and \begin{align*}B\end{align*} are points of tangency for \begin{align*}\bigodot C\end{align*} and \begin{align*}\bigodot D\end{align*}.

  1. Is \begin{align*}\triangle AEC \sim \triangle BED\end{align*}? Why?
  2. Find \begin{align*}CE\end{align*}.
  3. Find \begin{align*}BE\end{align*}.
  4. Find \begin{align*}ED\end{align*}.
  5. Find \begin{align*}BC\end{align*} and \begin{align*}AD\end{align*}.

\begin{align*}\bigodot A\end{align*} is inscribed in \begin{align*}BDFH\end{align*}.

  1. Find the perimeter of \begin{align*}BDFH\end{align*}.
  2. What type of quadrilateral is \begin{align*}BDFH\end{align*}? How do you know?
  3. Draw a circle inscribed in a square. If the radius of the circle is 5, what is the perimeter of the square?
  4. Can a circle be inscribed in a rectangle? If so, draw it. If not, explain.
  5. Draw a triangle with two sides tangent to a circle, but the third side is not.
  6. Can a circle be inscribed in an obtuse triangle? If so, draw it. If not, explain.
  7. Fill in the blanks in the proof of the Two Tangents Theorem.

Given: \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{CB}\end{align*} with points of tangency at \begin{align*}A\end{align*} and \begin{align*}C\end{align*}. \begin{align*}\overline{AD}\end{align*} and \begin{align*}\overline{DC}\end{align*} are radii.

Prove: \begin{align*}\overline{AB} \cong \overline{CB}\end{align*}

Statement Reason
1. 1.
2. \begin{align*}\overline{AD} \cong \overline{DC}\end{align*} 2.
3. \begin{align*}\overline{DA} \perp \overline{AB}\end{align*} and \begin{align*}\overline{DC} \perp \overline{CB}\end{align*} 3.
4. 4. Definition of perpendicular lines
5. 5. Connecting two existing points
6. \begin{align*}\triangle ADB\end{align*} and \begin{align*}\triangle DCB\end{align*} are right triangles 6.
7. \begin{align*}\overline{DB} \cong \overline{DB}\end{align*} 7.
8. \begin{align*}\triangle ABD \cong \triangle CBD\end{align*} 8.
9. \begin{align*}\overline{AB} \cong \overline{CB}\end{align*} 9.
  1. Fill in the blanks, using the proof from #21.
    1. \begin{align*}ABCD\end{align*} is a _____________ (type of quadrilateral).
    2. The line that connects the ___________ and the external point \begin{align*}B\end{align*} __________ \begin{align*}\angle ABC\end{align*}.
  2. Points \begin{align*}A, \ B,\end{align*} and \begin{align*}C\end{align*} are points of tangency for the three tangent circles. Explain why \begin{align*}\overline{AT} \cong \overline{BT} \cong \overline{CT}\end{align*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.2. 

Resources

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Vocabulary

circle

The set of all points that are the same distance away from a specific point, called the center.

diameter

A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.

point of tangency

The point where the tangent line touches the circle.

radius

The distance from the center to the outer rim of a circle.

Tangent

The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle.

Tangent to a Circle Theorem

A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.

Two Tangent Theorem

The Two-Tangent Theorem states that if two tangent segments are drawn to one circle from the same external point, then they are congruent.

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