Question 1: In a circle, a diameter AB subtends a right angle at point C on the circumference. If angle ABC measures 45°, what is the measure of angle ACB?

Answer: 45

Question 2: Construct a triangle within a circle where one side is a diameter and the opposite angle is 90°. Use the grid to draw and label the triangle.

Answer: Student draws a right-angled triangle inscribed in a circle with diameter as hypotenuse.

Question 3: A chord in a circle creates a 90° angle at a point on the circumference. Explain why this confirms the circle theorem about angles in a semicircle.

Answer: Because the angle subtended by a diameter at the circumference is 90°, confirming the circle theorem.

Question 4: Plot the graph of y=2x on the provided grid. Identify the points where the line intersects the circle centered at the origin with radius 5.

Answer: Points at (x, y) where y=2x and x² + y²=25; solutions are approximately (2.24, 4.48) and (-2.24, -4.48).

Question 5: In a circle with diameter XY, point Z lies on the circle such that ∠XZY measures 90°. If ∠XYZ is 30°, find the measure of ∠YZX.

Answer: 60

Question 6: A semicircular arc has endpoints P and Q on the circle. Point R on the arc creates ∠PRQ = 90°. Describe how this relates to the circle theorem.

Answer: Because any angle subtended by a diameter in a circle is 90°, confirming the theorem.

Question 7: Construct a diameter and mark a point on the circumference such that the inscribed angle is 90°. Use the grid to illustrate.

Answer: Construct diameter AB, then choose any point C on the circle's arc opposite AB, forming a right angle at C.

Question 8: A problem involves a circle with radius 7. A triangle is inscribed with one vertex at the circle's edge such that the angle at that vertex is 90°. Find the length of the hypotenuse if the other two vertices are on the circle.

Answer: Hypotenuse is 14, because diameter is 2×radius, and the right triangle inscribed with hypotenuse as diameter has length 14.

Question 9: Identify the common mistake students make when applying the circle theorem for angles in a semicircle, and explain how to avoid it.

Answer: Incorrectly assuming the angle is always 90°, instead of recognizing it's only 90° when subtended by a diameter. To avoid this, always check if the side is a diameter.

Question 10: A circle has a diameter AB. Points C and D are on the circle such that ∠ACD = 90° and ∠ABD = 90°. Determine the possible positions of C and D relative to the diameter.

Answer: Points C and D must lie on the circle’s arc opposite the diameter, each forming right angles with the endpoints A and B, respectively.