Question

There are two parallel chords measuring 16 cm and 12 cm, both situated on the same side of the center of a circle. The space between the two chords is 2 cm. What is the radius of the circle?

• A. 8 cm
• B. 10 cm
• C. 12 cm
• D. 15 cm

Answer 1

Answer:

B. 10 cm

Let the radius be (R), and the distance of the chords from the center be (x) and (x+2) (since both are on the same side and 2 cm apart).

---

Step 1: Use chord formula

For a chord of length (l):
[
l = 2\sqrt{R^2 - d^2}
]

---

Step 2: Apply for 16 cm chord

[
16 = 2\sqrt{R^2 - x^2}
]
[
8 = \sqrt{R^2 - x^2}
]
[
R^2 - x^2 = 64 \quad ...(1)
]

---

Step 3: Apply for 12 cm chord

[
12 = 2\sqrt{R^2 - (x+2)^2}
]
[
6 = \sqrt{R^2 - (x+2)^2}
]
[
R^2 - (x+2)^2 = 36 \quad ...(2)
]

---

Step 4: Subtract (2) from (1)

[
(R^2 - x^2) - (R^2 - (x+2)^2) = 64 - 36
]

[
(x+2)^2 - x^2 = 28
]

Expand:
[
x^2 + 4x + 4 - x^2 = 28
]

[
4x + 4 = 28
]

[
4x = 24
]

[
x = 6
]

---

Step 5: Find radius

Use (1):
[
R^2 - 36 = 64
]

[
R^2 = 100
]

[
R = 10
]

---

✅ Final Answer:

[
\boxed{10\ \text{cm}}
]