Two chords of length 20 cm and 24 cm are drawn perpendicular to each other in a circle of radius 15 cm. What is the distance be tween the points of intersection of these chords (in cm) from the centre of the circle?

Options

A. `sqrt(114)`
B. `sqrt(182)`
C. `sqrt(246)`
D. `sqrt(218)`

Answer

To solve the problem, we need to find the distance from the center of the circle to the point of intersection of the two chords. Let's break this down step by step.

### Step 1: Understand the Geometry
We have two chords AB and CD in a circle with radius 15 cm. Chord AB has a length of 20 cm, and chord CD has a length of 24 cm. The chords intersect at point P and are perpendicular to each other.

### Step 2: Find the Distances from the Center to Each Chord
To find the distance from the center of the circle (O) to each chord, we can use the formula for the distance (d) from the center of the circle to a chord:

\[ d = \sqrt{r^2 - \left(\frac{l}{2}\right)^2} \]

where \( r \) is the radius of the circle and \( l \) is the length of the chord.

### Step 3: Calculate the Distance from the Center to Chord AB
For chord AB (length = 20 cm):
- Half of the chord length = \( \frac{20}{2} = 10 \) cm
- Using the formula:

\[ d_{AB} = \sqrt{15^2 - 10^2} = \sqrt{225 - 100} = \sqrt{125} = 5\sqrt{5} \text{ cm} \]

### Step 4: Calculate the Distance from the Center to Chord CD
For chord CD (length = 24 cm):
- Half of the chord length = \( \frac{24}{2} = 12 \) cm
- Using the formula:

\[ d_{CD} = \sqrt{15^2 - 12^2} = \sqrt{225 - 144} = \sqrt{81} = 9 \text{ cm} \]

### Step 5: Determine the Coordinates of Points A, B, C, and D
Assuming the center O is at the origin (0, 0):
- Let chord AB be horizontal, with points A and B at coordinates:
- A(-10, \( d_{AB} \)) = (-10, \( 5\sqrt{5} \))
- B(10, \( d_{AB} \)) = (10, \( 5\sqrt{5} \))

- Let chord CD be vertical, with points C and D at coordinates:
- C(0, \( d_{CD} \)) = (0, 9)
- D(0, -d_{CD}) = (0, -9)

### Step 6: Find the Coordinates of Point P (Intersection of Chords)
The intersection point P of the two chords is where they meet:
- Since chord AB is horizontal and chord CD is vertical, the coordinates of point P are:
- P(0, \( 5\sqrt{5} \))

### Step 7: Calculate the Distance from the Center to Point P
Finally, we need to find the distance from the center O(0, 0) to point P(0, \( 5\sqrt{5} \)):
- The distance is simply the y-coordinate of P:

\[ \text{Distance} = |5\sqrt{5}| \]

### Final Answer
The distance from the center of the circle to the point of intersection of the chords is \( 5\sqrt{5} \) cm.

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