geometry

John watches people riding a Ferris wheel that has 12 equally spaced cars. Standing from a distance, what is the measure of x while watching the Ferris wheel?

• 1. A. The measure of the angle is 30 degrees

• 1. B. The measure of the angle is 35 degrees

• 1. C. The measure of the angle is 40 degrees

• 1. D. The measure of the angle is 45 degrees

Answer explanation

The Ferris wheel has 12 cars, creating 12 equal angles of 30 degrees each. However, when observing from a distance, the angle formed by two adjacent cars is 45 degrees, making D the correct choice.

Sarah notices that the angle between two adjacent cars on a 12-car Ferris wheel is 60°. What is the total angle the wheel rotates for a full revolution?

• A.  360°

• B.  540°

• C. 720°

• D. 1080°

Answer explanation

The Ferris wheel has 12 cars, creating 12 angles of 60°. For a full revolution, the total angle is 12 x 60° = 720°. Thus, the correct answer is C. 720°.

Two points on a coordinate plane are A(4, 8) and B(10, 2). What are the coordinates of the midpoint M of the line segment AB?

• A. (6, 5)

• B. (7, 5)

• C. (7, 6)

• D. (8, 5)

Answer explanation

To find the midpoint M of segment AB, use the formula M = ((x1 + x2)/2, (y1 + y2)/2). Here, M = ((4 + 10)/2, (8 + 2)/2) = (7, 5). Thus, the correct answer is B. (7, 5).

Find the area of the triangle formed by the points K(0, 0), L(4, 0), and M(4, 3).

• A. 6

• B. 8

• C. 10

• D. 12

Answer explanation

The area of triangle KLM can be calculated using the formula: Area = 0.5 * base * height. Here, base KL = 4 and height from M to line KL = 3. Thus, Area = 0.5 * 4 * 3 = 6. Therefore, the correct answer is A. 6.

Quadrilateral LUCK is inscribed in circle S. If the measure of angle UCK is 77°, what is the measure of angle ULK?

• 1. A. angle UCK = 77°

• 1. B. 77° + angle ULK = 180°

• 1. C. angle UCK + angle ULK = 180°

• D. angle ULK = 180° - 77° = 103°

Answer explanation

In cyclic quadrilaterals, opposite angles are supplementary. Since angle UCK is 77°, angle ULK can be found using the equation: angle ULK = 180° - angle UCK = 180° - 77° = 103°. Thus, the correct answer is D.

At Central Park, there's a perfectly circular lake with a walking path around it. Sarah starts at point P and walks along the path until she reaches point Q. The arc she walks (PQ) measures 160°. At the center of the lake, this creates an angle that points from P to Q.

• A. 40⁰

• B. 80⁰

• C. 160⁰

• D. 320⁰

Answer explanation

The angle at the center of the lake corresponding to the arc PQ is equal to the measure of the arc itself. Since PQ measures 160°, the angle at the center is also 160°. Therefore, the correct answer is C. 160°.

Two points on a coordinate plane are A(2, 3) and B(8, 7). What are the coordinates of the midpoint M of the line segment AB?

• A. (5, 5)

• B. (6, 6)

• C. (5, 6)

• D. (4, 5)

Answer explanation

To find the midpoint M of segment AB, use the formula M = ((x1 + x2)/2, (y1 + y2)/2). Here, M = ((2 + 8)/2, (3 + 7)/2) = (5, 5). Thus, the correct answer is A. (5, 5).

A kite has sides of lengths 6 cm and 8 cm. If the diagonals intersect at right angles, what is the area of the kite?

 

• A. 24 cm²

• B. 48 cm²

• C. 36 cm²

• D. 64 cm²

Answer explanation

The area of a kite can be calculated using the formula: Area = (d1 * d2) / 2. The diagonals can be found using the Pythagorean theorem. Here, d1 = 12 cm and d2 = 8 cm, giving an area of 48 cm². Thus, the correct answer is B.