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What is Drawing a Pair of Tangents to a Circle Inclined at a Given Angle?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Drawing a pair of tangents to a circle inclined at a given angle means constructing two straight lines that touch the circle at exactly one point each, and these two lines meet each other at a specific angle outside the circle. Imagine two roads meeting at a corner, and both roads just 'kiss' a circular park.
Simple Example
Quick Example
Think about a round 'roti' (chapati) on a plate. If you place two rulers on the plate such that each ruler just touches the edge of the roti at one point, and these two rulers meet outside the roti to form an angle, that's what we're doing. The 'given angle' is the angle formed where the two rulers meet.
Worked Example
Step-by-Step
Let's say we need to draw two tangents to a circle of radius 3 cm, such that the tangents are inclined at an angle of 60 degrees to each other.
1. First, draw a circle with center O and radius 3 cm.
---2. We know the angle between the two tangents (60 degrees). The angle between the radii to the points of contact and the angle between the tangents add up to 180 degrees. So, the angle between the two radii will be 180 - 60 = 120 degrees.
---3. Draw any radius OA. From O, draw another radius OB such that angle AOB is 120 degrees.
---4. At point A, draw a line perpendicular to OA. This will be our first tangent.
---5. At point B, draw a line perpendicular to OB. This will be our second tangent.
---6. These two perpendicular lines will meet at a point, let's call it P. The angle APB will be 60 degrees.
---Answer: The two lines PA and PB are the required tangents inclined at 60 degrees.
Why It Matters
Understanding tangents helps engineers design smooth curves for roads and railway tracks, ensuring vehicles turn safely. In computer graphics, it's used to make objects look realistic and move smoothly. Even in robotics, it helps robots navigate around circular obstacles efficiently.
Common Mistakes
MISTAKE: Drawing the tangents so they cut through the circle, instead of just touching it. | CORRECTION: Remember, a tangent only touches the circle at one single point, never crossing inside.
MISTAKE: Confusing the angle between the tangents with the angle between the radii. | CORRECTION: The angle between the tangents and the angle between the radii (to the points of contact) are supplementary, meaning they add up to 180 degrees.
MISTAKE: Not drawing the radii to the points of contact perpendicular to the tangents. | CORRECTION: Always remember that the radius drawn to the point of tangency is always perpendicular to the tangent line at that point.
Practice Questions
Try It Yourself
QUESTION: If the angle between two tangents drawn to a circle is 70 degrees, what is the angle between the radii to the points of contact? | ANSWER: 110 degrees
QUESTION: Draw two tangents to a circle of radius 4 cm such that they are inclined at an angle of 90 degrees to each other. What kind of shape is formed by the center, the two points of contact, and the intersection point of the tangents? | ANSWER: A square
QUESTION: A circle has a radius of 5 cm. If two tangents are drawn from an external point P, and the distance from the center to P is 13 cm, find the length of each tangent. (Hint: Use Pythagoras theorem). | ANSWER: 12 cm
MCQ
Quick Quiz
If two tangents to a circle are inclined at 45 degrees, what is the angle formed by the two radii at the center, connecting to the points where the tangents touch?
45 degrees
90 degrees
135 degrees
180 degrees
The Correct Answer Is:
C
The angle between the tangents and the angle between the radii to the points of contact are supplementary. So, 180 - 45 = 135 degrees.
Real World Connection
In the Real World
Imagine a drone delivering a package in a crowded city like Bengaluru. The drone's path might need to smoothly go around a tall circular water tank. The drone's control system uses the concept of tangents to calculate the best path to just 'graze' the tank without hitting it, keeping the delivery smooth and efficient. This is crucial in logistics and autonomous vehicle navigation.
Key Vocabulary
Key Terms
TANGENT: A 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 | CIRCLE: A round plane figure whose boundary (the circumference) consists of points equidistant from a fixed center | INCLINED ANGLE: The angle formed where two lines meet | PERPENDICULAR: Two lines or segments that meet at a 90-degree angle
What's Next
What to Learn Next
Great job understanding tangents! Next, you can explore 'Properties of Tangents to a Circle' to learn more about their amazing characteristics and how they relate to the circle's radius. This will help you solve even more complex geometry problems!
