Math resources Geometry

Circle theorems

Tangent of a circle

# Tangent of a circle

Here you will learn about the circle theorems involving tangents of a circle, including their application, proof, and using them to solve more difficult problems.

Students will first learn about the tangent of a circle as part of geometry in high school.

## What is the tangent of a circle?

A tangent of a circle is a straight line that touches the circumference of the circle at a single point. This point is called the point of tangency.

• Diagram 1 – The angle between a tangent and radius is 90 degrees.
• Diagram 2 – Tangents which meet at the same point are equal in length.

In diagram 1 above, the tangent meets the circle at point A (the point of tangency), which is perpendicular to the radius of the circle at that point .

In diagram 2, two tangents meet the circle at two different points (B and D) and they intersect at point A.

If the points B and D are linked by a chord, AB and AD are the same length, so ABD is an isosceles triangle.

If the points B and D join to the center of the circle C, they form a kite ABCD. This means that you have the two circle theorems:

Isosceles triangle

Kite

### Key parts of a circle needed for these theorems

• The radius of a circle is the distance from the center to the circumference of the circle. The radius is half of the diameter.
• The center of the circle is a point that is located in the middle of a circle.
• The circumference of the circle is the distance around the edge of the circle.

### Proving that if two tangents meet, they are the same length

To be able to prove this theorem, you do not need to know any other circle theorem. You just need to be confident with angles in a triangle. You also need to understand congruence.

## Common Core State Standards

How does this relate to high school math?

• High school: Geometry – Circles (HS.G.C.A.4)
Construct a tangent line from a point outside a given circle to the circle.

## How to use the tangent theorems

In order to use the tangent of a circle:

1. Locate the key parts of the circle for the theorem.
2. Use other angle facts to determine the remaining angle(s) made with the tangent.
3. Use the tangent theorem to state the other missing angle.

## Tangent of a circle examples

### Example 1: standard diagram

Points A, B, and C are on the circumference of a circle with point O as the center. ( This can also be called circle O.) \, DE is a tangent at point A. Calculate the size of angle BAD.

1. Locate the key parts of the circle for the theorem.

Here you have:

• Angle BCA=52^{\circ}
• AC is the diameter
• DE is a tangent

2Use other angle facts to determine the remaining angle(s) made with the tangent.

As AC is a diameter and the angle in a semicircle is 90^{\circ}, angle ABC=90^{\circ}. As angles in a triangle total 180^{\circ},

\begin{aligned} \text{ Angle } CAB&=180-(90+52) \\\\ &=38^{\circ} \end{aligned}

3Use the tangent theorem to state the other missing angle.

As the angle between the tangent and the radius is 90^{\circ}, you can now calculate angle BAD\text{:}

### Example 2: angles in the same segment

A, B, C, and D are points on the circumference of a circle with center O. \, AC and BD intersect at point G. \, EF is a tangent at point C and is parallel to BD. Calculate the size of angle BCF.

Locate the key parts of the circle for the theorem.

Use other angle facts to determine the remaining angle(s) made with the tangent.

Use the tangent theorem to state the other missing angle.

### Example 3: angles at the center

A circle with center O has three points on the circumference, A, B, and C. The tangent DE passes through point C. Calculate the size of angle BCE.

Locate the key parts of the circle for the theorem.

Use other angle facts to determine the remaining angle(s) made with the tangent.

Use the tangent theorem to state the other missing angle.

### Example 4: tangent of a circle

B, C, and D are points on the circumference of a circle with center O. \, AE and AF are tangents to the circle. Calculate the size of angle DBF.

Locate the key parts of the circle for the theorem.

Use other angle facts to determine the remaining angle(s) made with the tangent.

Use the tangent theorem to state the other missing angle.

### Example 5: alternate segment theorem

The triangle ABC is inscribed in a circle with center O. \, DE is a tangent at point A. Calculate the size of the angle OAC.

Locate the key parts of the circle for the theorem.

Use other angle facts to determine the remaining angle(s) made with the tangent.

Use the tangent theorem to state the other missing angle.

### Example 6: complex diagram

ABCD is an arrowhead inscribed inside a circle with center C. The two tangents EF and GH meet at external point P. Calculate the size of angle FPG.

Locate the key parts of the circle for the theorem.

Use other angle facts to determine the remaining angle(s) made with the tangent.

Use the tangent theorem to state the other missing angle.

### Teaching tips for tangent of a circle

• Provide worksheets with real-world examples where tangents are encountered, such as the point where a road meets a circular roundabout or the point where a billiard ball strikes the cushion of a table. Relating abstract concepts to tangible situations enhances understanding.

• Start with simple practice questions and gradually increase complexity. This approach enhances critical thinking and problem-solving skills.

• Highlight connections between tangents and other mathematical concepts, such as the Pythagorean theorem (Pythagoras’ theorem), similar triangles, and trigonometry.

### Easy mistakes to make

• Misrepresenting the angle between the tangent and the radius
Either through miscalculation or an assumption, the angle between the tangent and the radius is not 90^{\circ} because “it doesn’t look like it”, it must be proven.

• Misusing the alternate segment theorem
There are cases when the Alternate Segment Theorem is used to describe the angle at the tangent or the angle in the alternate segment at the circumference but neither is true.

Take Example 5 above. The angle OAC is assumed to be equal to 56^{\circ} whereas angle CAE is equal to 56^{\circ}.

• Thinking the angle is double or half the opposing angle
The kite that is formed when two tangents meet has two angles of 90^{\circ} and 90^{\circ} because they meet the radius at 90^{\circ}.

The other two angles are assumed to be double or half whereas they should total 180^{\circ} (it is a unique case for the cyclic quadrilateral).

In the example below, the angle COB is correctly equal to 144^{\circ} as 72\times{2}=144^{\circ}. The angle CAB is incorrectly equal to 144\div{2}=72^{\circ}.

This is incorrect because angles in a quadrilateral should total 360^{\circ}, whereas shape ABOC has a total of 90+90+144+72=396^{\circ}. The only case when this is correct is when ABOC is a square.

• Circle theorems
• Central angle
• Chords of a circle
• Subtended
• Circle chord theorems

### Practice tangent of a circle questions

1. The right triangle ABC is inscribed in the circle with center O. The line DE is a tangent of the circle with the point of contact at A. Angle BAD=78^{\circ}. Calculate the size of the angle ACB.

12^{\circ}

78^{\circ}

90^{\circ}

68^{\circ}
• BAC=90-78=12^{\circ} \, ( the tangent meets the radius at 90^{\circ})
• ABC=90^{\circ} \, ( angles in a semicircle are 90^{\circ})
• ACB=180-(90+12)=78^{\circ} \, ( angles in a triangle sum to 180^{\circ})

2. A, B, C, and D are points on the circumference of a circle with center O. \, AC and BD are endpoints of two perpendicular lines with AC passing through the center of the circle. EF is a tangent to the circle at point C. Calculate the size of the angle BCF.

68^{\circ}

44^{\circ}

22^{\circ}

56^{\circ}
• DBC=68^{\circ} \, (BD is parallel to EF as BD and AC are perpendicular)
• BGC=90^{\circ} \, ( chord of a triangle)
• BCG=180-(90+68)=22^{\circ}
• 90-22=68^{\circ} \, ( the tangent meets the radius at 90^{\circ})

3. The triangle ABC is inscribed into a circle with center O. The line DE is a tangent at the point C. The angle BCE= \theta. Calculate the size of angle BCE.

61^{\circ}

29^{\circ}

45^{\circ}

90^{\circ}
• OCA=29^{\circ} \, ( triangle OAC is isosceles as OA=OC)
• ABC=90^{\circ} \, ( angles in a semicircle)
• OCB=90-29=61^{\circ}
• BCE=90-61=29^{\circ} \, ( the tangent meets the radius at 90^{\circ})

4. A circle with center O has three points on the circumference, B, C, and D. The lines AE and AF are tangents to the circle at points B and C. The tangents meet at the point A. Angle BAC=40^{\circ}. Calculate the size of the angle ODC.

40^{\circ}

80^{\circ}

60^{\circ}

55^{\circ}
• OAC=40\div{2}=20^{\circ} \, (OA bisects the angle BAC)
• ACO=90^{\circ} \, ( angles in a semicircle)
• AOC=180-(90+20)=70^{\circ} \, ( angles in a triangle)
• OCD is an isosceles triangle as OC=OD
• ODC=(180-70)\div{2}=55^{\circ} \, ( angles in an isosceles triangle)

5. A circle with center O has three points on the circumference, A, B, and C. The line DE is a tangent to the circle at the point A. Angle BAD=78^{\circ}. Calculate the size of the angle ABC.

86^{\circ}

4^{\circ}

90^{\circ}

88^{\circ}
• EAC=4^{\circ} \, ( alternate segment theorem)
• OAC=90^{\circ} \, ( angles in a semicircle are 90^{\circ})
• OAC=90-4=86^{\circ}

6. A, B, and D are points on the circle with center C. The tangents EF and GH intersect at the point P at an angle of 112^{\circ}. Calculate the size of angle BAD.

112^{\circ}

68^{\circ}

56^{\circ}

34^{\circ}
• BPD=180-112=68^{\circ} \, ( angles on a straight line total 180^{\circ})
• CDP=CBP=90^{\circ} \, ( the tangent meets the radius at 90^{\circ})
• BCD=360-(90+90+68)=112^{\circ} \, ( angles in a quadrilateral total 360^{\circ})
• BAD=112\div{2}=56^{\circ} \, ( angle at the center is twice the angle at the circumference)

## Tangent of a circle FAQs

What is a tangent to a circle?

A tangent to a circle is a straight line that touches the circle at exactly one point, without crossing it. This single point where the circle touches the line is called the point of tangency.

What is the tangent-secant theorem?

The Tangent-Secant Theorem states that if a tangent and a secant are drawn from a point outside a circle, then the square of the length of the tangent segment is equal to the product of the lengths of the whole secant segment and its external segment.

How do you find the length of a tangent segment from a point to a circle?

If you know the distance from the point to the center of the circle (d) and the radius of the circle (r), the length of the tangent segment can be found using the Pythagorean theorem: Length of tangent =\sqrt{d^2-r^2}

How do you find the point of tangency?

To find the point of tangency, you need the coordinates of the point and the equation of the circle. The point of tangency will satisfy both the equation of the circle and the equation of the tangent line.

What is the difference between the tangent of a circle and a chord of a circle?

A tangent is a straight line that touches a circle at exactly one point. A chord is a line segment with both endpoints on the circumference of the circle.

## Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.