Complementary angles are angles whose measures add up to 90°. Let's call the measures of the two complementary angles 3x and 2x.

Thus, the equation can be written as:

3x + 2x = 90
5x = 90

Dividing both sides by 5 to solve for x:

x = 18

Now we substitute x back into 2x, which represents the measure of the smaller angle:

2(18) = 36

So, the measure of the smaller angle is 36°. The correct answer is A) 36°.

Since angle A is 90 degrees, triangle ABC is a right-angled triangle with the right angle at vertex A. We can use the Pythagorean Theorem (a² + b² = c²) to find the length of BC, where a and b are the two legs of the right-angled triangle and c is the hypotenuse.

In triangle ABC, a = AB = 5, and b = AC = 12. We have to find the length of the hypotenuse BC.

5² + 12² = c²
25 + 144 = c²
169 = c²

Taking the square root of both sides:

c = 13

So, the length of BC is 13. The correct answer is C) 13.

Let r be the radius of the larger circle. The area of the annulus is the difference between the areas of the larger and the smaller circles.

area_annulus = π(r²) - π(6²)
42π = π(r²) - 36π

Adding 36π to both sides and factoring out π, we get:

78π = π(r²)

Dividing both sides by π:

78 = r²

Taking the square root of both sides:

r = √78

So, the radius of the larger circle is √78. The correct answer is B) √78.

To find the circumference of a circle, we can use the formula:

Circumference = π × diameter

Given the diameter of the circle is 10 inches, we have:

Circumference = π × 10

So, the circumference of the circle is 10π inches. The correct answer is B) 10π.

To find the length of the diagonal of a square, we can use the Pythagorean theorem, as the diagonal divides the square into two congruent right triangles. Let d represent the length of the diagonal and s represent the side length of the square.

In each right triangle, the legs are of length s, and the hypotenuse is of length d. According to the Pythagorean theorem:

s² + s² = d²

Given that the side length of the square is 8, we have:

8² + 8² = d²
64 + 64 = d²
128 = d²

Taking the square root of both sides:

d = √128 = 8√2

So, the length of the diagonal is 8√2. The correct answer is B) 8√2.