Answers
Answer 1
1) The area of a circle circumscribed about a square is 307.7 cm².
2.a.) The angle ACB is 39 degrees.°.
2b.) The value of x is 5.42.
How to determine the area of a circle?We shall find the radius to determine the area of a circle.
First, find the side length of the square:
Since the perimeter of the square = 56 cm, then, each side of the square is 56 cm / 4 = 14 cm.
Next, find the diagonal of the square, using the Pythagorean theorem:
Diagonal = the diameter of the circumscribed circle.
Diagonal² = side length² + side length²
= 14 cm² + 14 cm²
= 196 cm² + 196 cm²
= 392 cm²
Take the square root of both sides:
Diagonal = √392 cm ≈ 19.80 cm (rounded to two decimal places)
Then, the radius of the circle which is half the diagonal:
Radius = Diagonal / 2 ≈ 19.80 cm / 2 ≈ 9.90 cm (rounded to two decimal places)
Finally, compute the area of the circle using the formula:
Area = π * Radius²
Area = 3.14 * (9.90 cm)²
Area ≈ 307.7 cm² (rounded to two decimal places)
Therefore, the area of the circle that is circumscribed about a square with a perimeter of 56 cm is 307.7 cm².
2. a) We use the property of angles in a circle to solve for angle ACB: an angle inscribed in a circle is half the measure of its intercepted arc.
Given that arc AB has a measure of 78°, we can find angle ACB as follows:
Angle ACB = 1/2 * arc AB
= 1/2 * 78°
= 39°
Therefore, the angle ACB is 39 degrees.
2b.) To solve for the value of x, we use the information that the angle ADB = (3x - 12)⁴.
Given that angle ADB is (3x - 12)⁴, we can equate it to the measure of the intercepted arc AB, which is 78°:
(3x - 12)⁴ = 78
Solve the equation for x, by taking the fourth root of both sides:
∛∛((3x - 12)⁴) = ∛∛78
Simplify,
3x - 12 = ∛(78)
Isolate x by adding 12 to both sides:
3x - 12 + 12 = ∛(78) + 12
3x = ∛(78) + 12
Finally, divide both sides by 3:
x = (∛(78) + 12) / 3
x = (4.27 +12) / 3
x = 5.42
So, x is 5.42
Therefore,
1) The area of the circle is 154 cm².
2a.) Angle ACB is equal to 102°.
2b.) The value of x is 5.42
Learn more about the area of a circle at brainly.com/question/12374325
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