Mathematics 10 — Exercise Review · Q5 (iv)

Full question

For the given z, verify the displayed identity.

Step-by-step solution

1. First write z and its conjugate. Only the imaginary sign changes.

   $$(4\times i+3,3-4\times i)$$

2. Read a, the real coordinate.

   $$a=3$$

3. Read b, the coefficient of i, including its sign.

   $$b=4$$

4. Use |z| = √(a² + b²) and substitute both coordinates.

   $$\sqrt{3^2+4^2}$$

5. Square the real coordinate.

   $$3^2=9$$

6. Square the imaginary coordinate; a negative number squared is positive.

   $$4^2=16$$

7. Add the two squared values.

   $$9+16=25$$

8. Take the nonnegative square root of this sum.

   $$\sqrt{25}=5$$

9. Repeat the modulus calculation for the comparison number.

10. Read a, the real coordinate.

    $$a=3$$

11. Read b, the coefficient of i, including its sign.

    $$b=-4$$

12. Use |z| = √(a² + b²) and substitute both coordinates.

    $$\sqrt{3^2+{\left(-4\right)}^2}$$

13. Square the real coordinate.

    $$3^2=9$$

14. Square the imaginary coordinate; a negative number squared is positive.

    $${\left(-4\right)}^2=16$$

15. Add the two squared values.

    $$9+16=25$$

16. Take the nonnegative square root of this sum.

    $$\sqrt{25}=5$$

17. Sign changes disappear when squared, so the distances match.

    $$5=5$$

Answer

The final answer is shown below.