Mathematics 10 — Exercise 9.1 · Q6

Full question

A fountain has radius 3 m. A pole is 7 m from its centre. Find each tangent rope length.

Step-by-step solution

1. Join the centre O to the contact point T and the external point P. OT is perpendicular to PT.

2. OP is the hypotenuse. Pythagoras gives PT²=OP²−OT². Take the positive square root.

3. Substitute the given numbers into the formula.

   $$\sqrt{7^2-3^2}$$

4. Calculate this operation before combining the remaining terms.

   $$7^2=49$$

5. Calculate this operation before combining the remaining terms.

   $$3^2=9$$

6. Calculate this operation before combining the remaining terms.

   $$\left(-1\right)\times 9=-9$$

7. Calculate this operation before combining the remaining terms.

   $$49-9=40$$

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

   $$40=4\times 10$$

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

   $$\sqrt{40}=2\times \sqrt{10}$$

10. Calculate this operation before combining the remaining terms.

    $$\sqrt{40}=2\times \sqrt{10}$$

11. The required result is shown below. Unit: m

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

Answer

The final answer is shown below.