Question 3

The length of a chord depends on its distance from the center and the circle's radius.

Step 1 — Using the chord length formula

Let the circle's radius be $\mathbf{r}$. Let the distance of chord CD from the center be $\mathbf{d}$. The distance of chord AB from the center is $\mathbf{2d}$. The formula for chord length is $2\sqrt{\text{radius}^2 - \text{distance}^2}$.

Length of chord CD: $CD = 2\sqrt{r^2 - d^2}$

Length of chord AB: $AB = 2\sqrt{r^2 - (2d)^2}$ $= 2\sqrt{r^2 - 4d^2}$

Now, let's check if $CD = 2AB$. $2AB = 2 \times (2\sqrt{r^2 - 4d^2})$ $= 4\sqrt{r^2 - 4d^2}$

We need to see if $2\sqrt{r^2 - d^2} = 4\sqrt{r^2 - 4d^2}$. This equality is not true in general. It only holds for specific values of $\mathbf{r}$ and $\mathbf{d}$. Therefore, we cannot conclude $CD = 2AB$.

$\boxed{\text{CD is not generally equal to 2AB}}$

Step 2 — Checking with an example

Let's use specific values. Let the radius $\mathbf{r = 5}$ cm. Let the distance of CD from the center $\mathbf{d = 2}$ cm. The distance of AB from the center is $\mathbf{2d = 4}$ cm.

Calculate the length of chord CD. $CD = 2\sqrt{5^2 - 2^2}$ $= 2\sqrt{25 - 4}$ $= 2\sqrt{21}$

Calculate the length of chord AB. $AB = 2\sqrt{5^2 - 4^2}$ $= 2\sqrt{25 - 16}$ $= 2\sqrt{9}$ $= 2 \times 3$ $= 6$

Now, let's find $2AB$. $2AB = 2 \times 6$ $= 12$

We compare $CD$ and $2AB$. $CD = 2\sqrt{21}$ and $2AB = 12$. Since $2\sqrt{21} \approx 9.16$, it is not equal to $12$. So, $CD \neq 2AB$.

$\boxed{\text{CD} \neq 2AB \text{ in this example}}$

Answer

(i) No, we cannot conclude that CD = 2 AB. (ii) The length of a chord is given by $2\sqrt{r^2 - d^2}$. (iii) This formula shows a non-linear relationship with distance $\mathbf{d}$.