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Tangent Lines - Worksheet 9 - 10th Grade

Hard

Apply properties of tangent lines to circles (Hard)

23 problems · 20-25 minutes · Skills: circles, tangents, geometry

About This Worksheet

This hard-level worksheet contains 23 problems designed for 10th Grade students practicing tangent lines. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

Tangent line properties have direct applications in physics (tangential velocity, circular motion) and engineering (gear systems, pulleys, track design). They also introduce students to limit-based thinking — a tangent line touches a circle at exactly one point — which foreshadows calculus concepts.

  • Prove and apply the theorem that a tangent is perpendicular to the radius at the point of tangency
  • Apply the Two-Tangent Theorem to find equal tangent segment lengths
  • Solve problems involving angles and segments formed by tangents and secants
  • Calculate the length of a tangent segment from an external point using the Pythagorean Theorem
  • Apply the secant-tangent angle theorem to find intercepted arc measures

How to Solve: Worked Example

Example Problem: From a point 29 from the centre of a circle of radius 20, find the tangent segment length.
Solution Steps:
  1. Here the hypotenuse (29) and one leg (the radius, 20) are known.
  2. The tangent segment is the other leg: √(29² - 20²).
  3. = √(841 - 400) = √441.
  4. = 21.
Answer: 21
Tip: Subtract when finding a leg, add when finding the hypotenuse. Identify which one you need first.
1
From a point 65 from the centre of a circle of radius 33, find the length of the tangent segment.
2
From a point 30 from the centre of a circle of radius 18, find the length of the tangent segment.
3
From a point 61 from the centre of a circle of radius 11, find the length of the tangent segment.
4
From a point 5 from the centre of a circle of radius 3, find the length of the tangent segment.
5
From a point 26 from the centre of a circle of radius 10, find the length of the tangent segment.
6
From a point 5 from the centre of a circle of radius 3, find the length of the tangent segment.
7
From a point 73 from the centre of a circle of radius 48, find the length of the tangent segment.
8
From a point 65 from the centre of a circle of radius 33, find the length of the tangent segment.
9
From a point 37 from the centre of a circle of radius 12, find the length of the tangent segment.
10
From a point 26 from the centre of a circle of radius 10, find the length of the tangent segment.
11
From a point 13 from the centre of a circle of radius 5, find the length of the tangent segment.
12
From a point 26 from the centre of a circle of radius 10, find the length of the tangent segment.
13
From a point 37 from the centre of a circle of radius 12, find the length of the tangent segment.
14
From a point 25 from the centre of a circle of radius 15, find the length of the tangent segment.
15
From a point 5 from the centre of a circle of radius 3, find the length of the tangent segment.
16
From a point 5 from the centre of a circle of radius 3, find the length of the tangent segment.
17
From a point 39 from the centre of a circle of radius 15, find the length of the tangent segment.
18
From a point 17 from the centre of a circle of radius 8, find the length of the tangent segment.
19
From a point 30 from the centre of a circle of radius 18, find the length of the tangent segment.
20
From a point 20 from the centre of a circle of radius 12, find the length of the tangent segment.
21
From a point 53 from the centre of a circle of radius 28, find the length of the tangent segment.
22
From a point 73 from the centre of a circle of radius 48, find the length of the tangent segment.
23
From a point 73 from the centre of a circle of radius 48, find the length of the tangent segment.

Answer Key

1. 562. 243. 604. 45. 246. 47. 558. 569. 3510. 2411. 1212. 2413. 3514. 2015. 416. 417. 3618. 1519. 2420. 1621. 4522. 5523. 55

Frequently Asked Questions

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Click the "Download PDF / Print" button at the top of the worksheet. You can save it as a PDF or print it directly. The answer key will be on a separate page.

How do I find worksheets at different difficulty levels?

Each subtopic has 10 worksheets: Worksheets 1-3 are Easy, 4-7 are Medium, and 8-10 are Hard. Start with Easy worksheets to build confidence, then progress to harder ones.

What skills does this circles worksheet cover?

This worksheet focuses on circles, tangents, geometry. It is designed for 10th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

Can I use this worksheet in my classroom?

Yes. All worksheets on K12Worksheets are free for classroom use. You can print copies for your entire class. With 10 worksheets per subtopic at three difficulty levels, you have plenty of options for differentiated instruction and individualized practice.

How many problems are on this worksheet?

This worksheet contains 23 problems. It typically takes 20-25 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.