Question 1: In a circle, a diameter AB is drawn. An angle inscribed in the semicircle with vertex C on the circle is measured as 90°. If the length of the diameter AB is 12 cm, what is the length of the chord AC if the angle at C is 90°? (Hint: Use Pythagoras’ theorem).

Answer: 8.49

Question 2: Construct a semicircle with diameter 10 cm. Mark a point on the circle to form a right-angled triangle inscribed in the semicircle. State the measure of the inscribed angle at that point.

Answer: 90°

Question 3: If the radius of a circle is 7 cm, what is the measure of the inscribed angle subtended by a diameter?

Answer: 90°

Question 4: Plot a circle with radius 5 cm and draw a diameter. On the circle, draw a point that forms a right angle with the endpoints of the diameter. What is the measure of this angle?

Answer: 90°

Question 1: A semicircular pond has a diameter of 14 meters. A boat is anchored at a point on the semicircle such that the angle between the boat and the two ends of the diameter is 90°. What is the maximum distance the boat could be from the midpoint of the diameter? (Hint: Use the properties of right triangles inscribed in a semicircle).

Answer: 7√2 meters

Question 2: A circle of radius 9 cm has a diameter AB. A point C on the circle creates a right-angled triangle ABC inscribed in the circle. If the length of AC is 12 cm, find the length of BC.

Answer: 15 cm

Question 3: In a circle, point D is on the circumference. The diameter AB is 16 cm, and point D forms a right triangle with points A and B. If AD measures 10 cm, find the length of BD.

Answer: 12.8 cm

Question 1: A bridge is designed with a semi-circular arch. If the diameter of the arch is 20 meters, what is the angle formed between the arch and a horizontal line at the midpoint of the arch? Explain your reasoning.

Answer: 90° because the angle inscribed in a semicircle is a right angle.

Question 2: A solar panel is installed on a semi-circular roof with a diameter of 8 meters. If the panel is installed at a point on the roof that forms a right angle with the edges of the roof, what is the significance of this angle in terms of sunlight exposure?

Answer: The right angle ensures optimal sunlight absorption, as it relates to the angle inscribed in the semicircle being 90°.

Question 1: Construct a circle with a radius of 6 cm. Inscribe a triangle such that one side is the diameter and the other two points are on the circle. Show that the angles at the endpoints of the diameter are 90°, and determine the measure of the angle at the third point if the triangle is isosceles.

Answer: Angles at endpoints of diameter are 90°; the angle at the third point depends on the lengths but is less than 90° unless special conditions apply.

Question 2: Given a circle with an inscribed triangle where two sides are equal, and one of the angles opposite the equal sides is 90°, prove that the triangle is a right isosceles triangle and find the measures of the other two angles.

Answer: The other two angles are 45° each, making the triangle right-angled and isosceles.

Question 1: True or False: Any inscribed angle inscribed in a semicircle measures 90°. Explain your answer.

Answer: True

Question 2: Draw a semicircle with diameter 15 cm. Mark a point on the circumference such that the inscribed angle is not 90°. What can be said about this inscribed angle?

Answer: It is not 90°, as only angles inscribed in a semicircle are 90°.

Question 1: A student claims that an inscribed angle in a semicircle measures 180°. Is this correct? Explain your reasoning and correct if necessary.

Answer: Incorrect; an inscribed angle in a semicircle measures 90°, not 180°.