Q. Two circles intersect at points A and B. If the radius of the first circle is 5 cm and the second is 3 cm, what is the maximum distance between the centers of the circles?
Q. Two circles intersect at points A and B. What is the relationship between the angles ∠AOB and ∠APB, where O is the center of one circle and P is the center of the other?
A.
∠AOB = ∠APB
B.
∠AOB = 2∠APB
C.
∠AOB = ½∠APB
D.
∠AOB + ∠APB = 180 degrees
Solution
The angle ∠AOB is twice the angle ∠APB because of the inscribed angle theorem, which states that the angle at the center is twice the angle at the circumference.
Q. Two parallel lines are intersected by a transversal. If one of the corresponding angles is 75 degrees, what is the measure of the other corresponding angle?
A.
75 degrees
B.
105 degrees
C.
90 degrees
D.
60 degrees
Solution
Corresponding angles are equal, so the other corresponding angle is also 75 degrees.
Q. Two parallel lines are intersected by a transversal. If one of the corresponding angles is 65 degrees, what is the measure of the other corresponding angle?
A.
65 degrees
B.
115 degrees
C.
180 degrees
D.
90 degrees
Solution
Corresponding angles are equal when two parallel lines are cut by a transversal.
Q. Two triangles are similar if their corresponding angles are equal. If triangle DEF is similar to triangle XYZ, and angle D = 30 degrees, what is the measure of angle X?
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
120 degrees
Solution
Since the triangles are similar, corresponding angles are equal. Therefore, angle X = angle D = 30 degrees.
Q. Two triangles are similar if their corresponding angles are equal. If triangle DEF is similar to triangle XYZ, and angle D = 50 degrees, what is the measure of angle X?
A.
50 degrees
B.
60 degrees
C.
70 degrees
D.
80 degrees
Solution
Since the triangles are similar, corresponding angles are equal. Therefore, angle X = angle D = 50 degrees.
Q. Two triangles are similar with a ratio of their corresponding sides as 3:5. If the area of the smaller triangle is 27 cm², what is the area of the larger triangle?
A.
45 cm²
B.
75 cm²
C.
60 cm²
D.
50 cm²
Solution
Area ratio = (side ratio)² = (3/5)² = 9/25. Let area of larger triangle be A. 27/A = 9/25, A = 27 * (25/9) = 75 cm².
Q. Two triangles are similar. If the lengths of the sides of the first triangle are 3, 4, and 5 units, what are the lengths of the corresponding sides of the second triangle if the shortest side is 6 units?
A.
6, 8, 10
B.
9, 12, 15
C.
12, 16, 20
D.
15, 20, 25
Solution
The ratio of the sides is 6/3 = 2. Therefore, the corresponding sides are 6*2, 4*2, and 5*2, which are 6, 8, and 10 units.
Q. Two triangles are similar. If the sides of the first triangle are 3 cm, 4 cm, and 5 cm, what is the area of the second triangle if its longest side is 10 cm?
A.
40 cm²
B.
20 cm²
C.
30 cm²
D.
50 cm²
Solution
The ratio of the sides is 10/5 = 2. Area ratio = (2)² = 4. Area of first triangle = 6 cm². Area of second triangle = 6 * 4 = 24 cm².
Q. Two triangles are similar. If the sides of the first triangle are 3 cm, 4 cm, and 5 cm, what are the corresponding sides of the second triangle if the shortest side is 6 cm?
A.
8 cm, 10 cm, 12 cm
B.
9 cm, 12 cm, 15 cm
C.
6 cm, 8 cm, 10 cm
D.
12 cm, 16 cm, 20 cm
Solution
The ratio of similarity is 6/3 = 2. Therefore, the sides are 3*2 = 6 cm, 4*2 = 8 cm, 5*2 = 10 cm.
Q. Two triangles are similar. If the sides of the first triangle are 3 cm, 4 cm, and 5 cm, what is the length of the corresponding side in the second triangle if its longest side is 10 cm?
A.
6 cm
B.
8 cm
C.
10 cm
D.
12 cm
Solution
The ratio of the sides is 10/5 = 2. Therefore, the corresponding side is 4 * 2 = 8 cm.
Q. Two triangles are similar. If the sides of the first triangle are 3 cm, 4 cm, and 5 cm, what is the length of the longest side of the second triangle if its shortest side is 6 cm?
A.
8 cm
B.
10 cm
C.
12 cm
D.
9 cm
Solution
The ratio of the sides is 6/3 = 2. Therefore, the longest side is 5 * 2 = 10 cm.
Q. What are the roots of the polynomial x^2 + 2x - 8?
A.
-4 and 2
B.
4 and -2
C.
2 and -4
D.
0 and -8
Solution
To find the roots, we can factor the polynomial: x^2 + 2x - 8 = (x + 4)(x - 2). Setting each factor to zero gives us x + 4 = 0 or x - 2 = 0, so the roots are x = -4 and x = 2.
Q. What are the roots of the polynomial x^2 - 5x + 6?
A.
1 and 6
B.
2 and 3
C.
3 and 2
D.
5 and 0
Solution
To find the roots, we can factor the polynomial: x^2 - 5x + 6 = (x - 2)(x - 3). Setting each factor to zero gives us x - 2 = 0 or x - 3 = 0, so the roots are x = 2 and x = 3.
