Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 10 chapters and video availability01 · Read and understand
What you will learn
- Apply the tangent-radius perpendicular relation and Pythagoras.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Algebraic equations, ratios, angle and area facts, and basic coordinate geometry.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A circle has radius 5 cm. A point P is 13 cm from its center O. A tangent from P touches at T. Find PT.
Why this math matters
Convert a tangency condition into a right-triangle measurement. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use Euclidean geometry and the angle, parallelism, similarity, or congruence conditions stated; a sketch alone does not establish them.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Use the right angle at a tangent point
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A circle has radius 5 cm. A point P is 13 cm from its center O. A tangent from P touches at T. Find PT.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
OT ⟂ PT at the tangent point
The radius to a tangency point is perpendicular to the tangent line.
Develop the calculation
PT²+5²=13²
Triangle OTP is right with hypotenuse OP.
Check and interpret
PT=√144=12 cm
The positive root gives the geometric length.
The result
PT=√144=12 cm
The positive root gives the geometric length.
Common mistakes to catch
- The center-to-external-point length is the hypotenuse.
- An external point is required for two distinct tangent segments.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the length of the other tangent from the same P to the circle?
Show a hint
Tangents from one external point have equal lengths.
Reveal answer and explanation
12 cm
The two right triangles share OP and have equal radii, giving equal remaining legs.
Practice 2
Would the same construction work if OP=3 cm with radius five?
Show a hint
The point would be inside the circle.
Reveal answer and explanation
No real tangent from P exists
Every line through an interior point cuts the circle rather than touching it at just one point.
Take the idea with you
Convert a tangency condition into a right-triangle measurement.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
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