High School Geometry - Chords, Secants and Tangents

Introduction

• A chord is a line segment joining any two points on the boundary of a circle.
• A line intersecting a circle at two distinct points is called a Secant.
• A line intersecting or touching a circle at only one point is called its tangent.

Important Points

• The measure of an angle formed by two chords intersecting inside a circle is equal to half the sum of the measure of the arcs intercepted by the angle and its vertical angle counterpart.

In the figure shown above, $m\angle AMB=m\angle CMD=\frac{1}{2}\left(m\stackrel{⏜}{CD}+m\stackrel{⏜}{AB}\right)$

• The measure of an angle formed by a tangent and a chord meeting at the point of tangency is half the measure of the intercepted arc.

In the figure shown above, $m\angle BPQ=\frac{1}{2}m\stackrel{⏜}{QP}$ and $m\angle APQ=\frac{1}{2}m\stackrel{⏜}{QRP}$

• If two secants intersect outside a circle, then the measure of the angle formed is equal to half the positive difference of the measures of the intercepted arcs.

In the figure shown above, $m\angle APB=\frac{1}{2}\left|m\stackrel{⏜}{AB}-m\stackrel{⏜}{CD}\right|$

• If the two chords of a circle are equal in measure, then their corresponding minor arcs are equal in measure, and vice-versa.

In the figure above, chords AB and CD are equal in measure. Thus, $m\stackrel{⏜}{AB}=m\stackrel{⏜}{CD}$.

• If the diameter of a circle is perpendicular to a chord, then it bisects the chord and its arcs.

In the figure shown above, if the diameter CD is perpendicular to the chord AB, then AM = BM.

• If two chords of a circle are equal in measure, then they are equidistant from the center.

In the figure shown above, if chords AB and CD are equal, then OM = ON.

• If two chords intersect inside a circle, then the product of the segments of one chord equals the product of the segments of the other chord.

In the figure shown above, $a\times b=c\times d$

• If two secant segments intersect outside a circle, then the product of the secant segment with its external portion equals the product of the other secant segment with its external portion.

In the figure shown above, $PA\times PC=PB\times PD$

• If a tangent segment and a secant segment intersect outside a circle, then the square of the measure of the tangent segment equals the product of the measures of the secant segment and its external portion.

In the figure shown above, ${\left(PC\right)}^2=PB\times PD$

• The lengths of tangents drawn from an external point to a circle are equal.

In the figure shown above, $PB=PC$

Solved Examples

Example 1: In the figure shown $m\stackrel{⏜}{AB}=120°$ and $m\stackrel{⏜}{CD}=100°$. Find the measure of $x$.

Solution: $x=\frac{1}{2}\times \left(120°+100°\right)$ $=\frac{1}{2}\times 220°$ $=110°$

Example 2:  Find the measure of $\angle BPQ$ as shown in the figure if $m\stackrel{⏜}{PQ}=108°$.

Solution: $\angle BPQ=\frac{1}{2}m\stackrel{⏜}{PQ}$ $=\frac{1}{2}\times 108°$ $=54°$

Example 3: In the figure shown, $m\stackrel{⏜}{CD}=16°$ and $m\stackrel{⏜}{AB}=58°$. Find the measure of $\angle APB$.

Solution: $\angle APB=\frac{1}{2}\left|m\stackrel{⏜}{AB}-m\stackrel{⏜}{CD}\right|$ $=\frac{1}{2}\left|58°-16°\right|$ $=\frac{1}{2}\times 42°$ $=21°$

Example 4: Find the measure of the unknown length $x$ shown in the figure.

Solution: $10x=6\times 5$

$x=\frac{6\times 5}{10}$ $=3 \text{units}$

Example 5: In the figure shown, if $PC=5 \text{units}$ and $PB=10 \text{units}$, find the length of CD.

Solution: ${\left(PB\right)}^2=PC\times PD$

${\left(10\right)}^2=5\times PD$

$PD=\frac{100}{5}=20 \text{units}$

$PD=PC+CD$

$20=5+CD$

$CD=15 \text{units}$

Cheat Sheet

• The measure of an angle formed by two secants, by two tangents, or by a secant and tangent intersecting in the exterior of a circle is one-half the difference between the measures of the intercepted arcs.
• If two chords intersect inside a circle, then the product of the segments of one chord equals the product of the segments of the other chord.
• If two secant segments intersect outside a circle, then the product of the secant segment with its external portion equals the product of the other secant segment with its external portion.
• If a tangent segment and a secant segment intersect outside a circle, then the square of the measure of the tangent segment equals the product of the measures of the secant segment and its external portion.
• The tangent at any point of a circle is perpendicular to the radius through the point of contact.
• The lengths of tangents drawn from an external point to a circle are equal.

Blunder Areas

• Tangent to a circle is a special secant in which the two endpoints of its corresponding chord coincide.