The given figure shows two congruent circles with centres O and P respectively.Given that OP = 17 cm and OM = 15 cm, calculate the length of QR (in cm).

1. Break Down the Problem

We need to find the length of $QR$ created by the two congruent circles with centers $O$ and $P$. Given the following information:

• $OP = 17$ cm (the distance between the centers of the circles)
• $OM = 15$ cm (the radius of the circles, as both are congruent)

2. Relevant Concepts

Since both circles are congruent, it implies that the radius from each center to any point on the circle is the same. Thus:

• The radius $r$ of circle $O$ is $OM = 15$ cm.
• The radius $r$ of circle $P$ is also $PM$ (which is also $15$ cm since both circles are congruent).

To find $QR$, we can use the distance between the centers and the properties of circles to find the common chord length.

3. Analysis and Detail

Using the properties of intersecting circles, we can apply the following steps:

• The distance $d$ between the centers is $OP = 17$ cm.
• Each radius $r = 15$ cm.

We can find the length of $QR$ (the common chord) using the following formula:

$QR = 2 \sqrt{r^2 - \left(\frac{d}{2}\right)^2}$

Where $d$ is the distance between the centers of the circles. Substituting the known values:

1. Calculate $\frac{d}{2}$:

   $\frac{17}{2} = 8.5 \text{ cm}$

2. Compute $r^2 - \left(\frac{d}{2}\right)^2$:

   $QR = 2 \sqrt{15^2 - 8.5^2}$ $= 2 \sqrt{225 - 72.25}$ $= 2 \sqrt{152.75}$ $= 2 \times 12.35 \approx 24.70 \text{ cm}$

4. Verify and Summarize

The calculation appears to be sound, and after double-checking the inputs, we find that the calculations adhere to the properties of the circles. Thus, we can now present the final result.

Final Answer

The length of $QR$ is approximately $24.70$ cm.