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Summary

This tutorial explains how to determine the nearest and farthest distances from a point to a circle, covering three scenarios: when the point is inside, on, or outside the circle. The script demonstrates step-by-step how to use the distance formula and Pythagorean theorem to calculate these distances. It includes examples that illustrate the process for each case, showing how to apply mathematical principles to find the required distances, both for points inside and outside the circle. The tutorial is designed to help students understand these geometric concepts clearly through practical applications and explanations.

Takeaways

• 😀 A point inside a circle has a closest distance to the circle equal to the radius minus the distance from the point to the circle's center.
• 😀 The farthest distance for a point inside the circle is the radius plus the distance from the point to the circle's center.
• 😀 A point on the circle has no closest or farthest distance from the circle since it's already on the boundary.
• 😀 A point outside the circle has a closest distance equal to the distance between the point and the circle's center minus the radius.
• 😀 The farthest distance for a point outside the circle is the length of the tangent from the point to the circle, calculated using the Pythagorean theorem.
• 😀 To determine the closest and farthest distances from a point to a circle, first determine the position of the point relative to the circle (inside, on, or outside).
• 😀 If the substitution result into the circle's equation is less than zero, the point is inside the circle.
• 😀 If the substitution result equals zero, the point lies on the circle, meaning the closest and farthest distances are both zero.
• 😀 If the substitution result is greater than zero, the point lies outside the circle, and the distances can be calculated using the tangent line method for the farthest distance.
• 😀 The closest distance from a point inside the circle to the boundary is the difference between the radius and the distance from the point to the center, while the farthest distance is the sum of both.
• 😀 A specific example of determining distances from a point to a circle involved substituting coordinates into the circle’s equation to check the point’s location and calculating the distances using geometry and algebra.

Q & A

What are the three possible positions of a point relative to a circle?

The three possible positions of a point relative to a circle are: 1) Inside the circle, 2) On the circle, 3) Outside the circle.

How do you calculate the closest distance from a point inside the circle to the circle?

To calculate the closest distance from a point inside the circle to the circle, you subtract the distance from the point to the center of the circle (which is the radius) from the radius itself.

How is the farthest distance from a point inside the circle to the circle determined?

The farthest distance from a point inside the circle to the circle is determined by adding the distance from the point to the center of the circle to the radius.

What happens when a point lies on the circle? Does it have a closest and farthest distance?

When a point lies on the circle, it does not have a closest or farthest distance from the circle. Both distances are considered zero.

How do you determine the closest distance from a point outside the circle to the circle?

To determine the closest distance from a point outside the circle, you subtract the radius of the circle from the distance between the point and the center of the circle.

What is the farthest distance from a point outside the circle to the circle?

The farthest distance from a point outside the circle to the circle is the length of the tangent line from the point to the circle. This can be calculated using the Pythagorean theorem.

What formula is used to calculate the distance between two points in general?

The formula to calculate the distance between two points (x1, y1) and (x2, y2) is: √((x2 - x1)² + (y2 - y1)²).

How do you use substitution to determine the position of a point relative to a circle?

To determine the position of a point relative to a circle, you substitute the point's coordinates into the equation of the circle. If the result is less than zero, the point is inside the circle. If it equals zero, the point is on the circle. If it is greater than zero, the point is outside the circle.

How do you calculate the distance from a point to a circle when the point is outside the circle?

When a point is outside the circle, the distance to the circle is the difference between the distance from the point to the center of the circle and the radius of the circle.

What is the formula for calculating the tangent line length (farthest distance) from a point outside the circle to the circle?

The formula for calculating the length of the tangent line (farthest distance) is: √(distance between the point and center of the circle squared minus the radius squared).