A circle has a radius of 10 cm. What is the length of a tangent drawn from a poi

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Question: A circle has a radius of 10 cm. What is the length of a tangent drawn from a point 15 cm away from the center of the circle?

Options:

Correct Answer: √125

Solution:

Using the Pythagorean theorem, the length of the tangent is √(15^2 - 10^2) = √(225 - 100) = √125.

A circle has a radius of 10 cm. What is the length of a tangent drawn from a point 15 cm away from the center of the circle?

A circle has a radius of 10 cm. What is the length of a tangent drawn from a point 15 cm away from the center of the circle?

Step 1: Identify the radius of the circle, which is given as 10 cm.
Step 2: Identify the distance from the point to the center of the circle, which is given as 15 cm.
Step 3: Understand that the tangent line forms a right triangle with the radius and the line from the center to the point.
Step 4: Use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the distance from the point to the center) is equal to the sum of the squares of the other two sides (the radius and the tangent).
Step 5: Set up the equation: (length of tangent)^2 + (radius)^2 = (distance from point to center)^2.
Step 6: Substitute the known values into the equation: (length of tangent)^2 + (10 cm)^2 = (15 cm)^2.
Step 7: Calculate (10 cm)^2 = 100 and (15 cm)^2 = 225.
Step 8: Rewrite the equation: (length of tangent)^2 + 100 = 225.
Step 9: Subtract 100 from both sides: (length of tangent)^2 = 225 - 100.
Step 10: Calculate 225 - 100 = 125.
Step 11: Take the square root of both sides to find the length of the tangent: length of tangent = √125.
Step 12: Simplify √125 if needed, which is approximately 11.18 cm.

Pythagorean Theorem – The relationship between the lengths of the sides of a right triangle, where the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Circle Geometry – Understanding the properties of circles, including the relationship between the radius, tangent, and distance from the center.