ABC is the segment of a circle with center O. This segment is enclosed in a rectangle APQC. Given that AC=32cm and AP = 8 cm, calculate; (a) The radius of the circle. (b) The angle of AOC in both degrees and radians (c) The area of the shaded region.
(a) Why did the circle go on a diet? Because it wanted to be a smaller radius! To find the radius, we can use the fact that the diagonal of the rectangle APQC is the diameter of the circle. The length of the diagonal is given by Pythagoras' theorem, so let's calculate it:
APQC forms a right triangle with AC as the hypotenuse. Using Pythagoras' theorem, we have:
AC^2 = AP^2 + PC^2
Now let's plug in the values:
32^2 = 8^2 + PC^2
Simplifying the equation:
1024 = 64 + PC^2
Subtracting 64 from both sides:
PC^2 = 960
Now we can find the value of PC:
PC = √960
Finally, since the diagonal is the diameter of the circle, the radius is half of the diameter. So the radius of the circle is:
Radius = (1/2) * PC
(b) The angle of AOC. Well, isn't this angle just "acute"? Let's calculate it using some trigonometry! Since AP is the length between the center and the edge of the circle, and AC is the radius, we can use the sine function to find the angle AOC:
sin(AOC) = (AP/AC)
sin(AOC) = (8/32) = 1/4
Now, to find the angle AOC, we can use the inverse sine function:
AOC = arcsin(1/4)
(c) The area of the shaded region. Hmm, the shaded area seems a bit shady. To find the area, we need to subtract the area of the triangle APC from the area of the sector AOC. First, let's find the area of the triangle:
Area of triangle APC = (1/2) * AP * PC
And now let's find the area of the sector:
Area of sector AOC = (1/2) * radius^2 * AOC
Finally, we can find the area of the shaded region by subtracting the area of the triangle from the area of the sector. Remember, don't be too square about it!
Area of shaded region = Area of sector AOC - Area of triangle APC
(a) To find the radius of the circle, we can use the Pythagorean theorem in the right triangle AOC, where OA is the radius and AC is the hypotenuse:
AC^2 = OA^2 + OC^2
Substituting the given values into the equation:
32^2 = (OA)^2 + (OC)^2
1024 = OA^2 + OC^2
We also know that AP = 8 cm and AQ = 16 cm (since APQC is a rectangle). Therefore, the length of OC is 16 cm.
1024 = (OA)^2 + 16^2
1024 = (OA)^2 + 256
Subtracting 256 from both sides:
768 = (OA)^2
Taking the square root of both sides:
OA = √768
OA ≈ 27.7 cm
So, the radius of the circle is approximately 27.7 cm.
(b) To find the angle of AOC, we can use the property that an inscribed angle in a circle is one-half the measure of the central angle that subtends the same arc.
We have a right triangle AOC, so the central angle AOC is 90 degrees or π/2 radians.
Therefore, the angle of AOC is:
In degrees: 90 degrees
In radians: π/2 radians
(c) The area of the shaded region can be found by subtracting the area of the rectangle APQC from the area of the sector AOC.
The area of the rectangle APQC is given by length × width:
AP × QC = 8 cm × 32 cm = 256 cm²
The area of the sector AOC can be calculated using the formula:
Area of sector = (θ/360) × πr^2
Where θ is the central angle in degrees and r is the radius of the circle.
Substituting the given values into the formula:
Area of sector = (90/360) × π × (27.7 cm)^2
Area of sector ≈ 0.25 × π × (27.7 cm)^2
Area of sector ≈ 601.2 cm²
Therefore, the area of the shaded region is approximately 601.2 cm².
To answer these questions, we need to utilize the properties of a circle, as well as some basic geometry relationships.
(a) To find the radius of the circle, we can use the Pythagorean theorem. We know that AC is the diameter of the circle, so the radius is half of the diameter. Based on the given information, AC = 32 cm. Therefore, the radius (r) is 32/2 = 16 cm.
(b) To find the angle AOC, we need to use the fact that the radius of a circle is perpendicular to the tangent at the point of contact. In this case, we have a rectangle and the diagonals of a rectangle bisect each other.
The diagonals of rectangle APQC are AC and PC. Since AC is a diameter of the circle, it passes through the center O. So, angle AOC is half of the angle APC.
We have AP = 8 cm, and since PC is a side of the rectangle, it is equal to AP, so PC = 8 cm.
To find the angle APC, we can use trigonometry. The tangent of the angle can be calculated as tan(angle) = opposite/adjacent = AP/PC = 8/8 = 1. Taking the arctangent (inverse tangent) of both sides, we find that the angle APC is 45°.
Therefore, angle AOC is half of angle APC, which means it is 45°/2 = 22.5°.
To express this angle in radians, we need to convert it from degrees to radians. Since 180° equals π radians, we have 22.5° * π/180° = 0.3927 radians (rounded to four decimal places).
(c) To find the area of the shaded region, we can subtract the area of triangle AOC from the area of rectangle APQC.
First, let's find the area of triangle AOC. The area of a triangle can be calculated using the formula: area = (1/2) * base * height.
In triangle AOC, the base is AC (the diameter of the circle) and the height is OC (half the radius). So, the area of triangle AOC = (1/2) * AC * OC = (1/2) * 32 cm * 16 cm = 256 cm².
Next, the area of rectangle APQC is equal to the product of its length and width: area = AP * PC = 8 cm * 8 cm = 64 cm².
Therefore, the area of the shaded region is given by subtracting the area of triangle AOC from the area of rectangle APQC: shaded area = 64 cm² - 256 cm² = -192 cm². Note that the answer is negative because the triangle has a larger area than the rectangle in this case.
It's worth mentioning that the result of -192 cm² suggests that the given dimensions cannot form a rectangle enclosing a segment of a circle. Please check the information provided again to ensure accuracy.