1. Find the length of the chord. 2. The length of the arc of a circle is 22 cm, and its radius is 7 cm. Find the angle subtended by the arc at the center. 3. Two circles of radii 4... 1. Find the length of the chord. 2. The length of the arc of a circle is 22 cm, and its radius is 7 cm. Find the angle subtended by the arc at the center. 3. Two circles of radii 4 cm and 9 cm touch each other externally. Find the distance between their centers. 4. The radius of a circle is 5 cm. If a point is located 13 cm away from the center, determine if the point lies inside, on, or outside the circle. Also, find the length of tangent.
Understand the Problem
The image contains a series of questions related to circles, involving calculations of lengths, angles, and distances in relation to circles. It aims to assess the understanding of geometric properties and relationships among circles and arcs.
Answer
The angle is approximately $3.14$ radians, the distance between centers is $13$ cm, the point lies outside the circle, and the tangent length is $12$ cm.
Answer for screen readers
The angle subtended by the arc at the center is approximately ( 3.14 ) radians. The distance between the centers of the circles is ( 13 , \text{cm} ). The point is located outside the circle, and the length of the tangent is ( 12 , \text{cm} ).
Steps to Solve
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Finding the angle subtended by the arc at the center
The formula for the angle $\theta$ (in radians) subtended at the center by an arc is given by: $$ \theta = \frac{s}{r} $$ where ( s ) is the length of the arc and ( r ) is the radius.
Given ( s = 22 , \text{cm} ) and ( r = 7 , \text{cm} ), substitute these values into the formula: $$ \theta = \frac{22}{7} \approx 3.14 , \text{radians} $$
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Identifying the distance between the centers of the circles
If two circles touch each other externally, the distance ( d ) between their centers is the sum of their radii: $$ d = r_1 + r_2 $$ Given ( r_1 = 4 , \text{cm} ) and ( r_2 = 9 , \text{cm} ): $$ d = 4 + 9 = 13 , \text{cm} $$
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Determining the position of a point relative to the circle
For a circle of radius ( r = 5 , \text{cm} ) and a point located 13 cm from the center:
- If the distance from the center to the point is greater than the radius, the point lies outside the circle.
- If equal, it lies on the circle.
- If less, it lies inside.
In this case, ( 13 , \text{cm} > 5 , \text{cm} ), so the point lies outside the circle.
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Finding the length of tangent from a point outside the circle
The length of the tangent ( L ) from a point outside the circle can be found using: $$ L = \sqrt{d^2 - r^2} $$ where ( d = 13 , \text{cm} ) (distance from the center) and ( r = 5 , \text{cm} ): $$ L = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 , \text{cm} $$
The angle subtended by the arc at the center is approximately ( 3.14 ) radians. The distance between the centers of the circles is ( 13 , \text{cm} ). The point is located outside the circle, and the length of the tangent is ( 12 , \text{cm} ).
More Information
The angle can also be converted to degrees by multiplying with ( \frac{180}{\pi} ), which yields approximately ( 180^\circ ). The distance between centers in external tangents is crucial in geometry and applicable in various real-life situations like designing circular paths.
Tips
- Confusing the radius with the distance to a point outside the circle.
- Forgetting to convert radians to degrees when necessary.
- Using incorrect formulas for tangent lengths or distance between centers.
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