There are two chords AB and CD of length 10 cm and 24 cm vespectively both chords are opposed to centre If the distance between chords is 17 cm find radius

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Geometry
We have two chords AB and CD in a circle. The lengths of the chords are given as:
- Length of AB = 10 cm
- Length of CD = 24 cm

The distance between the two chords is given as 17 cm. We need to find the radius of the circle.

### Step 2: Find the Distances from the Center to the Chords
Let O be the center of the circle. The perpendicular distance from the center O to the chord AB is denoted as d1, and the perpendicular distance from O to the chord CD is denoted as d2.

Since the distance between the two chords is 17 cm, we can express this relationship as:
\[ d1 + d2 = 17 \text{ cm} \]

### Step 3: Use the Length of the Chords to Find Distances
The distance from the center of the circle to a chord can be found using the formula:
\[ d = \sqrt{r^2 - \left(\frac{l}{2}\right)^2} \]
where \( r \) is the radius of the circle and \( l \) is the length of the chord.

For chord AB (length = 10 cm):
\[ d1 = \sqrt{r^2 - \left(\frac{10}{2}\right)^2} = \sqrt{r^2 - 25} \]

For chord CD (length = 24 cm):
\[ d2 = \sqrt{r^2 - \left(\frac{24}{2}\right)^2} = \sqrt{r^2 - 144} \]

### Step 4: Set Up the Equation
From the earlier step, we have:
\[ d1 + d2 = 17 \]
Substituting the expressions for d1 and d2:
\[ \sqrt{r^2 - 25} + \sqrt{r^2 - 144} = 17 \]

### Step 5: Solve the Equation
To solve for \( r \), we will isolate one of the square roots:
1. Rearranging gives:
\[ \sqrt{r^2 - 144} = 17 - \sqrt{r^2 - 25} \]

2. Squaring both sides:
\[ r^2 - 144 = (17 - \sqrt{r^2 - 25})^2 \]
\[ r^2 - 144 = 289 - 34\sqrt{r^2 - 25} + (r^2 - 25) \]
\[ r^2 - 144 = r^2 + 264 - 34\sqrt{r^2 - 25} \]

3. Simplifying:
\[ -144 - 264 = -34\sqrt{r^2 - 25} \]
\[ -408 = -34\sqrt{r^2 - 25} \]
\[ \sqrt{r^2 - 25} = \frac{408}{34} = 12 \]

4. Squaring both sides again:
\[ r^2 - 25 = 144 \]
\[ r^2 = 169 \]
\[ r = \sqrt{169} = 13 \text{ cm} \]

### Conclusion
The radius of the circle is **13 cm**.

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