Mathematics 10 — Exercise Review · Q7

Full question

A branch forms a chord of length 20 m across a circular pond. A perpendicular rod from the centre to this chord is 10 m long. Find the radius of the pond.

Step-by-step solution

1. Draw the perpendicular from the centre O to the chord AB, meeting at M.

2. A perpendicular from the centre bisects a chord, so AM=MB=AB/2.

3. Divide the full chord length by 2.

   $$\mathrm{MB}=\frac{20}{2}$$

4. Calculate the half-chord length.

   $$\mathrm{MB}=10$$

5. In right triangle OMB the radius is the hypotenuse: r²=OM²+MB².

   $$r=\sqrt{{10}^2+{10}^2}$$

6. Calculate this operation before combining the remaining terms.

   $${10}^2=100$$

7. Calculate this operation before combining the remaining terms.

   $$100+100=200$$

8. Extract a perfect-square factor from under the root.

   $$200=100\times 2$$

9. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

   $$\sqrt{200}=10\times \sqrt{2}$$

10. Calculate this operation before combining the remaining terms.

    $$\sqrt{200}=10\times \sqrt{2}$$

11. Use the positive value because lengths and distances are nonnegative.

    $$10\times \sqrt{2}$$

Answer

The final answer is shown below.