Circle Tangent Practice Problems - Master All 8 Theorems

Practice circle tangent problems with step-by-step solutions. Master tangent theorems, radius relationships, and angle calculations through interactive exercises.

📚Master Circle Tangent Problems with These Essential Skills
  • Apply the perpendicular tangent-radius theorem to find missing angles
  • Calculate tangent lengths using the two-tangent equality property
  • Solve tangent-chord angle problems using inscribed angle relationships
  • Find unknown segments using the tangent-secant power theorem
  • Determine if quadrilaterals circumscribe circles using side sum conditions
  • Locate circle centers using triangle angle bisector intersections

Understanding Tangent to a circle

Complete explanation with examples

A tangent to a circle is a line that touches the circle at one point.

Tangent Theorem:

1) The tangent to the circle is perpendicular to the radius at the starting point

2) Every line perpendicular to the radius at its end is tangent to the circle

3) The angle between the tangent and any chord is equal to the circumferential angle that rests on that chord on the other side.

4) Two tangents to the circle that come out from the same point are equal to each other.

5) A segment that passes between the center of the circle and the point from which two tangents to the circle come out, cuts the angle between the tangents.

6) If from any point outside the circle, a tangent comes out and cuts the circle, then the product of the entire tangent on its outside is equal to the square of the tangent.

7) In the triangle that encloses the circle, the three bisectors of the angles of the triangle meet at a point in the center of the circle.

8) We can determine that a convex quadrilateral encloses a circle only if - the sum of two opposite sides in the square will be equal to the sum of the other two sides in the square.

1 - Tangent to the circle

Detailed explanation

Practice Tangent to a circle

Test your knowledge with 6 quizzes

Calculate the length of the arc marked in red given that the circumference is 12.

60°60°60°

Examples with solutions for Tangent to a circle

Step-by-step solutions included
Exercise #1

A point whose distance from the center of the circle is _______ than the radius, is outside the circle.

Step-by-Step Solution

Let's remember that the circle is actually the inner part of the circumference, meaning the enclosed area within the frame of the circumference.

Therefore, a point whose distance is greater than the center of the circle will necessarily be outside the circle.

Answer:

greater

Exercise #2

Where does a point need to be so that its distance from the center of the circle is the shortest?

Step-by-Step Solution

Let's remember that the circle is actually the inner part of the circumference, meaning the enclosed area within the frame of the circumference.

Therefore, a point whose distance is less than the radius from the center of the circle will necessarily be inside the circle.

Answer:

Inside

Exercise #3

In which of the circles is the point marked inside of the circle and not on the circumference?

Step-by-Step Solution

Let's remember that the circular line draws the shape of the circle, and the inner part is called a disk.

Therefore, in diagram B, the point is located in the inner part, meaning inside the disk.

Answer:

Video Solution
Exercise #4

Identify which diagram shows the radius of a circle:

Step-by-Step Solution

Remember that a radius is a line segment connecting the center of the circle to a point that lies on the circle itself.

In drawing A, the line doesn't touch any point on the circle itself.

In drawing B, the line doesn't pass through the center of the circle.

We can see that in drawing C, the line that extends from the center of the circle is indeed connected to a point on the circle itself.

Answer:

Video Solution
Exercise #5

Identify which diagram shows the radius of a circle:

Step-by-Step Solution

Remember that a radius is a line segment connecting the center of a circle to any point on the circle itself.

In drawing C we can see that the line coming from the center of the circle indeed connects to a point on the circle itself, while in the other drawings the lines don't touch any point on the circle.

Therefore, C is the correct drawing.

Answer:

Frequently Asked Questions

Everything you need to know about Tangent to a circle

What is a tangent to a circle and how do I identify it?

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A tangent to a circle is a line that touches the circle at exactly one point, called the point of tangency. Unlike secants that cross through the circle, tangents only touch the circumference and lie entirely outside the circle except at the contact point.

How do I prove that a line is tangent to a circle?

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To prove a line is tangent to a circle, show that it forms a 90° angle with the radius at the point where it touches the circle. This is the converse tangent theorem: any line perpendicular to a radius at its endpoint is tangent to the circle.

What are the 8 main tangent theorems I need to know?

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The 8 tangent theorems are: 1) Tangent perpendicular to radius, 2) Perpendicular to radius is tangent, 3) Tangent-chord angle equals inscribed angle, 4) Two tangents from same point are equal, 5) Line from center bisects tangent angle, 6) Tangent-secant power theorem, 7) Triangle angle bisectors meet at incenter, 8) Quadrilateral tangent condition (opposite sides sum).

How do I solve problems with two tangent lines from the same point?

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When two tangents come from the same external point, they are always equal in length. Additionally, the line connecting that point to the circle's center bisects the angle between the tangents, creating two congruent triangles.

What is the tangent-chord angle theorem and when do I use it?

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The tangent-chord angle theorem states that the angle between a tangent and any chord equals the inscribed angle on the opposite side of the chord. Use this when you need to find angles involving tangents and chords, especially in circle angle problems.

How does the power of a point theorem work with tangents?

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For a point outside a circle, if you draw a tangent and a secant from that point, then (tangent length)² = (external secant segment) × (whole secant length). This relationship helps solve for unknown lengths in tangent-secant configurations.

When does a quadrilateral circumscribe a circle?

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A convex quadrilateral circumscribes a circle if and only if the sum of opposite sides are equal. For quadrilateral ABCD: AB + CD = BC + AD. This condition ensures all four sides can be tangent to an inscribed circle.

What are common mistakes students make with tangent problems?

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Common mistakes include: confusing tangents with secants, forgetting the perpendicular relationship with radius, incorrectly identifying which inscribed angle corresponds to a tangent-chord angle, and not recognizing when two tangents from the same point create equal lengths and bisected angles.

Continue Your Math Journey

Master Tangent to a circle first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

CircleDistance from a chord to the center of a circleChords of a CircleCentral Angle in a CircleArcs in a CirclePerpendicular to a chord from the center of a circleInscribed angle in a circleCircuit ComponentsElements of the circumferenceArea of a circleThe Circumference of a CircleHow is the radius calculated using its circumference?

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