Using Circle Theorems: Focus on the Chord Bisector

The video tutorial focuses on chord bisectors and circle theorems, explaining how a perpendicular from the center of a circle bisects a chord. It covers the use of coordinate geometry to solve problems involving chord bisectors, including calculating gradients, finding equations of lines, and determining points of intersection. The tutorial demonstrates how to prove a line is a perpendicular bisector and verifies points on a circle. It concludes with a summary of the key concepts discussed.

What does a chord do to a circle?

• It divides the circle into two equal parts.

• It splits the circle into two segments.

• It forms a triangle inside the circle.

• It creates a tangent to the circle.

According to the circle theorem, what does a perpendicular from the center to a chord do?

• It forms a triangle.

• It divides the circle into quadrants.

• It bisects the chord.

• It creates a tangent.

What is the first step in solving problems involving chord bisectors using coordinate geometry?

• Finding the radius of the circle.

• Calculating the gradients.

• Drawing the circle.

• Identifying the center of the circle.

How can you prove that two lines are perpendicular using gradients?

• Their gradients are equal.

• Their gradients are negative reciprocals.

• Their gradients add up to zero.

• Their gradients are both positive.

In the example problem, what is the equation of the chord given?

• y = x + 5

• y - x = 5

• x + y = 7

• x - y = 5

What is the method used to find the intersection points of the chord and the circle?

• Substituting the chord equation into the circle equation.

• Using the Pythagorean theorem.

• Drawing a diagram.

• Using the quadratic formula.

How do you verify that a line is the perpendicular bisector of a chord?

• Check if it passes through the circle's center and the chord's midpoint.

• Confirm it divides the circle into equal parts.

• Ensure it is parallel to the chord.

• Verify it is tangent to the circle.

What is the first step to show that two points lie on a circle?

• Draw the circle.

• Substitute the points into the circle's equation.

• Calculate the distance between the points.

• Find the midpoint of the points.

What is the gradient of a line that is perpendicular to a line with a gradient of -1?

• 2

• 0

• 1

• -1

What is the final step to confirm that a perpendicular bisector passes through the circle's center?

• Calculate the distance from the bisector to the circle's edge.

• Verify the bisector's equation holds true for the center's coordinates.

• Check if the bisector is parallel to the chord.

• Ensure the bisector is tangent to the circle.