What is the Length of a Tangent? | Simple Explanation for Class 10

S6-SA1-0271

What is the Length of a Tangent from an External Point to a Circle?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The length of a tangent from an external point to a circle is the distance from that external point to the point where the tangent touches the circle. This length can be calculated using the Pythagorean theorem, as the tangent is always perpendicular to the radius at the point of contact.

Simple Example

Quick Example

Imagine you are flying a kite and the string is perfectly straight. If the kite is at an external point and the circle is a round pond, the length of the string from your hand (external point) to where it just touches the edge of the pond (tangent point) is what we are talking about. It's the shortest straight path.

Worked Example

Step-by-Step

Let's find the length of a tangent from an external point P to a circle. The radius of the circle is 5 cm and the distance from the center O to the external point P is 13 cm.

Step 1: Draw a diagram. Draw a circle with center O. Mark an external point P. Draw a tangent from P to the circle, touching at point T. Draw the radius OT.
---Step 2: Identify the right-angled triangle. The radius OT is perpendicular to the tangent PT at the point of contact T. So, triangle OTP is a right-angled triangle with the right angle at T.
---Step 3: Apply the Pythagorean theorem. In triangle OTP, OP is the hypotenuse. So, OP^2 = OT^2 + PT^2.
---Step 4: Substitute the given values. We have OP = 13 cm and OT (radius) = 5 cm. So, 13^2 = 5^2 + PT^2.
---Step 5: Calculate the squares. 169 = 25 + PT^2.
---Step 6: Isolate PT^2. PT^2 = 169 - 25 = 144.
---Step 7: Find PT. PT = sqrt(144) = 12 cm.

Answer: The length of the tangent PT is 12 cm.

Why It Matters

Understanding tangent lengths is crucial for designing circular objects and structures, from bicycle gears to satellite orbits. Engineers use this concept in robotics to calculate gripper movements and in space technology for planning spacecraft trajectories around planets. It's fundamental for careers in mechanical engineering and aerospace.

Common Mistakes

MISTAKE: Assuming the tangent doesn't form a right angle with the radius. | CORRECTION: Always remember that the radius drawn to the point of tangency is perpendicular to the tangent, forming a 90-degree angle.

MISTAKE: Confusing the hypotenuse in the Pythagorean theorem. | CORRECTION: The distance from the center to the external point (OP) is always the hypotenuse in the right-angled triangle formed (OTP).

MISTAKE: Forgetting to take the square root at the end. | CORRECTION: After calculating the square of the tangent length (PT^2), remember to take the square root to find the actual length (PT).

Practice Questions

Try It Yourself

QUESTION: A circle has a radius of 6 cm. An external point P is 10 cm away from the center of the circle. What is the length of the tangent from P to the circle? | ANSWER: 8 cm

QUESTION: The length of a tangent from a point P to a circle is 24 cm. If the distance from P to the center of the circle is 25 cm, find the radius of the circle. | ANSWER: 7 cm

QUESTION: Two circles with radii 5 cm and 3 cm have their centers 17 cm apart. Find the length of the common tangent if it touches both circles on the same side of the line joining their centers. (Hint: Form a rectangle and a right triangle by drawing a line parallel to the common tangent). | ANSWER: 15 cm

MCQ

Quick Quiz

If the radius of a circle is 8 cm and the distance from the center to an external point is 17 cm, what is the length of the tangent from that point to the circle?

9 cm

15 cm

19 cm

25 cm

The Correct Answer Is:

Using the Pythagorean theorem, 17^2 = 8^2 + tangent^2. So, 289 = 64 + tangent^2, which means tangent^2 = 225. Therefore, the tangent length is sqrt(225) = 15 cm.

Real World Connection

In the Real World

This concept helps engineers design the perfect curve for roads or railway tracks where they meet a circular roundabout. Think about how a train track (tangent) needs to smoothly connect to a circular loop. Also, in satellite communication, understanding tangent paths helps engineers calculate the line of sight from a ground station to a satellite orbiting Earth.

Key Vocabulary

Key Terms

TANGENT: A straight line that touches a circle at exactly one point | RADIUS: A line segment from the center of a circle to any point on its circumference | EXTERNAL POINT: A point located outside the circle | PYTHAGOREAN THEOREM: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a^2 + b^2 = c^2) | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle

What's Next

What to Learn Next

Next, you should explore properties of tangents, like the fact that tangents drawn from an external point to a circle are equal in length. This builds directly on what you've learned and will help you solve more complex geometry problems involving circles and tangents.