A circle has radius 5 cm. Three chords oflengths 6 cm, 8 cm and 10 cm are drawnwhich chord subtends the largest angle on itsminor arc?

1. Break Down the Problem

To find which chord subtends the largest angle on its minor arc, we will use the relationship between the length of the chord and the angle it subtends at the center of the circle. We will use the following steps:

• Find the angle subtended by each chord using the radius of the circle and the length of each respective chord.
• Compare the angles to determine which chord subtends the largest angle.

2. Relevant Concepts

The relationship between the chord length $c$, the radius $r$, and the angle $\theta$ (in radians) subtended by the chord at the center is given by the formula: $c = 2r \sin\left(\frac{\theta}{2}\right)$ We can rearrange this formula to find the angle $\theta$: $\sin\left(\frac{\theta}{2}\right) = \frac{c}{2r}$ $\frac{\theta}{2} = \arcsin\left(\frac{c}{2r}\right)$ Thus, $\theta = 2 \arcsin\left(\frac{c}{2r}\right)$

3. Analysis and Detail

Given:

• Radius $r = 5$ cm

We will calculate $\theta$ for each chord.

For the first chord (c = 6 cm): $\sin\left(\frac{\theta_1}{2}\right) = \frac{6}{2 \cdot 5} = \frac{6}{10} = 0.6$ $\frac{\theta_1}{2} = \arcsin(0.6)$ Calculating $\theta_1$: $\theta_1 = 2 \arcsin(0.6) \approx 2 \times 0.6435 \approx 1.287 \text{ radians}$

For the second chord (c = 8 cm): $\sin\left(\frac{\theta_2}{2}\right) = \frac{8}{2 \cdot 5} = \frac{8}{10} = 0.8$ $\frac{\theta_2}{2} = \arcsin(0.8)$ Calculating $\theta_2$: $\theta_2 = 2 \arcsin(0.8) \approx 2 \times 0.9273 \approx 1.854 \text{ radians}$

For the third chord (c = 10 cm): $\sin\left(\frac{\theta_3}{2}\right) = \frac{10}{2 \cdot 5} = \frac{10}{10} = 1$ $\frac{\theta_3}{2} = \arcsin(1)$ Calculating $\theta_3$: $\theta_3 = 2 \times \frac{\pi}{2} = \pi \text{ radians}$

4. Verify and Summarize

Now we have the angles:

• $\theta_1 \approx 1.287 \text{ radians}$
• $\theta_2 \approx 1.854 \text{ radians}$
• $\theta_3 = \pi \text{ radians} \approx 3.142 \text{ radians}$

Thus, the chord of length 10 cm subtends the largest angle at the center of the circle.

Final Answer

The chord of length 10 cm subtends the largest angle on its minor arc.