6.18: Tangent Lines

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

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.

$$\overleftrightarrow{BC}$$ is tangent at point $$B$$ if and only if $$\overleftrightarrow{BC}\perp \overline{AB}$$.

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.

$$\overline{BC}$$ and $$\overline{DC}$$ have $$C$$ as an endpoint and are tangent; $$\overline{BC}\cong \overline{DC}$$.

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?

Example $$\PageIndex{1}$$

Determine if the triangle below is a right triangle.

Solution

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

Does

\begin{aligned} 8^2+10^2&= (4\sqrt{10})^2? \\ 64+100&\neq 160\end{aligned}

$$\Delta ABC$$ is not a right triangle. From this, we also find that $$\overline{CB}$$ is not tangent to $$\bigodot A$$.

Example $$\PageIndex{2}$$

If $$D$$ and $$C$$ are the centers and $$AE$$ is tangent to both circles, find $$DC$$.

Solution

$$\overline{AE}\perp \overline{DE}$$ and $$\overline{AE}\perp \overline{AC}$$ and $$\Delta ABC \sim \Delta DBE$$ by AA Similarity.

To find $$DB$$, use the Pythagorean Theorem.

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

To find $$BC$$, use similar triangles.

$$\dfrac{5}{10}=\dfrac{BC}{26}\rightarrow BC=13. \: DC=DB+BC=26+13=39$$

Example $$\PageIndex{3}$$

$$\overline{CB}$$ is tangent to $$\bigodot A$$ at point $$B$$. Find $$AC$$. Reduce any radicals.

Solution

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

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

Example $$\PageIndex{4}$$

Using the answer from Example A above, find $$DC$$ in $$\bigodot A$$. Round your answer to the nearest hundredth.

Solution

\begin{aligned} DC&=AC−AD \\ DC&=\sqrt{89}−5 \approx 4.43 \end{aligned}

Example $$\PageIndex{5}$$

Find the perimeter of $$\Delta ABC$$.

Solution

$$AE=AD$$, $$EB=BF$$, and $$CF=CD$$. Therefore, the perimeter of

$$\Delta ABC=6+6+4+4+7+7=34$$.

$$\bigodot G$$ is inscribed in $$\Delta ABC$$. 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 $$\bigodot K$$.

Find the value of the indicated length(s) in $$\bigodot C$$. $$A$$ and $$B$$ are points of tangency. Simplify all radicals.

$$A$$ and $$B$$ are points of tangency for $$\bigodot C$$ and $$\bigodot D$$.

1. Is $$\Delta AEC \sim \Delta BED$$? Why?
2. Find $$CE$$.
3. Find $$BE$$.
4. Find $$ED$$.
5. Find $$BC$$ and $$AD$$.

$$\bigodot A$$ is inscribed in $$BDFH$$.

1. Find the perimeter of $$BDFH$$.
2. What type of quadrilateral is $$BDFH$$? 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: $$\overline{AB}$$ and $$\overline{CB}$$ with points of tangency at $$A$$ and $$C$$. $$\overline{AD}$$ and $$\overline{DC}$$ are radii.

Prove: $$\overline{AB}\cong \overline{CB}$$

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

Vocabulary

Term Definition
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.