Equal Chords and Their Distances from the Centre MCQs

Q1. A circle of radius 13 cm has one chord 5 cm from the centre and another 12 cm from the centre. The chord nearer the centre is longer; by how many cm does the longer chord exceed the shorter?

• A.14 cm
• B.24 cm
• C.7 cm
• D.10 cm

Answer: A. 14 cm

Reading the problem, 13 cm, 5 cm, 12 cm (math, chapter 'Circles'). What we must find: how many cm does the longer chord exceed the shorter?. From the chapter we use the relation: Nearer chord (distance 5) = 2×.√.(13^2-5^2) = 24. The arithmetic is: Nearer chord (distance 5) = 2×.√.(13^2-5^2) = 24 → farther = 10 → difference = 24 - 10 = 14 cm. That lands on option A) 14 cm. As for the others, option B) '24 cm' doesn't hold - That is the longer chord itself; the question wants the DIFFERENCE; option C) '7 cm' fails since Find each chord by Pythagoras, then subtract the lengths.

Q2. A circle of radius 17 cm has one chord 8 cm from the centre and another 15 cm from the centre. The chord nearer the centre is longer; by how many cm does the longer chord exceed the shorter?

• A.14 cm
• B.30 cm
• C.7 cm
• D.16 cm

Answer: A. 14 cm

Reading the problem, 17 cm, 8 cm, 15 cm (math, chapter 'Circles'). The unknown we need is how many cm does the longer chord exceed the shorter?. The principle that connects these is - Nearer chord (distance 8) = 2×.√.(17^2-8^2) = 30. Putting the numbers in: Nearer chord (distance 8) = 2×.√.(17^2-8^2) = 30 → farther = 16 → difference = 30 - 16 = 14 cm. That lands on option A) 14 cm. As for the others, option B) '30 cm' fails since That is the longer chord itself; the question wants the DIFFERENCE; option C) '7 cm' is wrong because Find each chord by Pythagoras, then subtract the lengths.

Q3. Two equal chords of a circle of radius 5 cm are each 3 cm from the centre. For each chord, by how many cm does its length exceed its distance from the centre?

• A.5 cm
• B.8 cm
• C.1 cm
• D.2 cm

Answer: A. 5 cm