Test on Circle Theorems and Chords

Question 1: In Example 1.6, calculating the length of PO, the Pythagorean theorem is applied as PO^2 = PQ^2 - OQ^2.

A. Ano

B. Ne

Explanation: According to the solution for Example 1.6, the Pythagorean theorem is applied as $PO^2 = 8^2 + OQ^2$, which corresponds to $PO^2 = PQ^2 + OQ^2$, not subtraction.

Question 2: Given a chord AB of a circle with center O, if OD is perpendicular to AB, AD is 12 cm, and the radius is 13 cm, the length of chord AB is 26 cm.

A. Ano

B. Ne

Explanation: In the provided Exercise 1.1, it is stated that OD is perpendicular to chord AB and AD = 12 cm. According to Theorem 1 (as applied in other examples like Example 1.7 and 1.6), a line from the center perpendicular to a chord bisects the chord. Therefore, D is the midpoint of AB, meaning AB = 2 * AD. Given AD = 12 cm, the length of AB is 2 * 12 cm = 24 cm. The radius of 13 cm is used to calculate OD, but not directly for AB's length if AD is known.

Question 3: In a circle with center O, OE is a radius. If OD = 15 cm and the radius of the circle is 22 cm, what is the length of DE?

A. 37 cm

B. 7 cm

C. 22 cm

D. 15 cm

Explanation: To calculate the length of DE, we use the fact that OE is a radius. We know that OE = OD + DE. Given that the radius is 22 cm, OE = 22 cm. Given OD = 15 cm. Therefore, 22 = 15 + DE. Subtracting 15 from 22 gives DE = 7 cm.

Question 4: In a circle with centre O, if AB is a chord, OD is perpendicular to AB, AD = 12 cm, and the radius of the circle is 13 cm, what is the length of OD?

A. 5 cm

B. 7 cm

C. 10 cm

D. 13 cm

Explanation: In the right-angled triangle ODA (since OD is perpendicular to AB), OA is the radius (hypotenuse) and AD is half the chord. Using the Pythagorean theorem, OA^2 = OD^2 + AD^2. We are given OA = 13 cm (radius) and AD = 12 cm. So, 13^2 = OD^2 + 12^2. This gives 169 = OD^2 + 144. Subtracting 144 from 169 yields OD^2 = 25. Therefore, OD = sqrt(25) = 5 cm. This method is consistent with how similar calculations (e.g., finding ML or radius PO) are performed using the Pythagorean theorem in the study materials.

Question 5: If a line drawn from the center of a circle is perpendicular to a chord, it divides the chord into two equal segments.

A. Ano

B. Ne

Explanation: According to Theorem 1, a line drawn from the center of the circle perpendicular to the chord bisects the chord. This means it divides the chord into two equal segments.