in circle p, chord cd is 12 inches long and measure of arc cd=90°, find the exact area of the segment bounded by chord cd and arc cd.

arc cd=90° ??

this makes no sense, since arc length is measured in linear units , not degrees

Are you saying that the arc cd subtends a 90° angle at the centre?

If so,
then r^2 + r^2 = 12^2
2r^2 = 144
r^2 = 72
r = 6√2

area of whole circle = 72π inches^2
area of 1/4 = 18π

area of triangle with 12 as the hypotenuse
= (1/2)(√72)(√72) = 36

area of segment = 18π - 36 inches^2
or appr 20.55 inches^2

To find the exact area of the segment bounded by chord CD and arc CD in circle P, we can use the formula for the area of a segment. The formula for the area of a segment is:

Area of Segment = Area of Sector - Area of Triangle

Here's how you can find the area:

1. Start by finding the radius of the circle. Since we have information about the chord, we can use the formula for the length of a chord in terms of the radius:
Length of Chord = 2 * Radius * sin(theta/2)

In this case, the length of chord CD is given as 12 inches, and the angle of the arc CD is given as 90 degrees:
12 = 2 * Radius * sin(90/2)

2. Solve for the radius:
12 = 2 * Radius * sin(45)
12 = 2 * Radius * √(2)/2
12 = Radius * √(2)
Radius = 12 / √(2)
Radius = 6√(2)

3. Now that we have the radius, we can find the area of the sector using the formula:
Area of Sector = (θ/360) * π * r^2

Since the angle of the arc is 90 degrees (θ = 90), we can substitute the values:
Area of Sector = (90/360) * π * (6√(2))^2
Area of Sector = (1/4) * π * 36 * 2
Area of Sector = 9π

4. Next, we need to find the area of the triangle. The triangle is formed by the chord and two radii of the circle. Since the angle between the radii is 90 degrees, the triangle is a right triangle. We can use the formula for the area of a right triangle:
Area of Triangle = (1/2) * Base * Height

In this case, the base is the length of the chord (12 inches), and the height is the radius (6√(2) inches):
Area of Triangle = (1/2) * 12 * 6√(2)
Area of Triangle = 36√(2)

5. Finally, we can find the area of the segment by subtracting the area of the triangle from the area of the sector:
Area of Segment = Area of Sector - Area of Triangle
Area of Segment = 9π - 36√(2)

Therefore, the exact area of the segment bounded by chord CD and arc CD in circle P is 9π - 36√(2) square units.