Two mutually perpendicular chords AB and CD meet at a point P inside the circle such that AP = 6 cms, PB = 4 cms, and DP = 3 cms. What is the area of the circle?

Step 1: Understand the problem We are given a circle with two mutually perpendicular chords AB and CD that intersect at point P. We know the lengths of AP, PB, and DP, and we need to find the area of the circle.

Step 2: Draw a diagram It's always helpful to draw a diagram for geometry problems. Draw a circle and mark the chords AB and CD intersecting at point P. Mark the lengths given in the problem.

Step 3: Use the properties of chords In a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. This means that APPB = CPPD.

Step 4: Substitute the given values We know that AP = 6 cms, PB = 4 cms, and DP = 3 cms. Substitute these values into the equation from step 3 to get 64 = CP3. Solve for CP to get CP = 8 cms.

Step 5: Find the diameter of the circle The diameter of the circle is the sum of the lengths of CP and DP, which is 8 + 3 = 11 cms.

Step 6: Find the area of the circle The area of a circle is given by the formula πr², where r is the radius of the circle. The radius is half the diameter, so r = 11/2 = 5.5 cms. Substitute this into the formula to get the area = π*(5.5)² = 94.985 cms².