Similarity Right Triangles and Trigonometry Practice Questions
Common Core (US) · Common Core Geometry · 146 free MCQs with instant results and detailed explanations.
146
Total
45
Easy
76
Medium
25
Hard
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Sample Questions from Similarity Right Triangles and Trigonometry
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Q1
Easy
In a right triangle, one of the angles is 30 degrees. If the length of the side opposite this angle is 5 units, what is the length of the hypotenuse?
A.
10 units
B.
5 units
C.
8.66 units
D.
6 units
Show Answer & Explanation
Correct Answer: A
In a 30-60-90 triangle, the ratio of the lengths of the sides is 1:√3:2. If the opposite side is 5 units (1 part), the hypotenuse is 10 units (2 parts).
Q2
Easy
If triangle ABC is similar to triangle DEF, with the ratio of their areas being 16:25, what is the ratio of their corresponding side lengths?
A.
4:5
B.
2:5
C.
16:25
D.
3:5
Show Answer & Explanation
Correct Answer: A
The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding side lengths. If the area ratio is 16:25, the side length ratio is √(16:25) = 4:5.
Q3
Easy
In a right triangle, if one angle measures 30 degrees, what is the ratio of the lengths of the side opposite the 30-degree angle to the hypotenuse?
A.
1:2
B.
√3:2
C.
1:√3
D.
2:1
Show Answer & Explanation
Correct Answer: A
In a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, giving a ratio of 1:2.
Q4
Medium
In triangle ABC, angle A is 90 degrees, and angle B is 30 degrees. If the length of side AC is 10 cm, what is the length of side AB?
A.
5 cm
B.
10 cm
C.
15 cm
D.
20 cm
Show Answer & Explanation
Correct Answer: A
In a 30-60-90 triangle, the sides are in the ratio 1:√3:2. Here, AB (the side opposite the 30-degree angle) is half of AC (10 cm), which gives us 5 cm.
Q5
Medium
A ladder is leaning against a wall, forming a right triangle with the ground. If the foot of the ladder is 6 feet from the wall and the ladder is 10 feet long, what is the height at which the ladder touches the wall?
A.
6 feet
B.
8 feet
C.
10 feet
D.
4 feet
Show Answer & Explanation
Correct Answer: B
Using the Pythagorean theorem, we have 6^2 + height^2 = 10^2. Solving gives height = 8 feet.
Q6
Medium
In two similar triangles, the perimeters are in the ratio 3:5. If the shorter triangle has a perimeter of 30 cm, what is the perimeter of the larger triangle?
A.
40 cm
B.
50 cm
C.
60 cm
D.
70 cm
Show Answer & Explanation
Correct Answer: B
If the ratio of perimeters is 3:5, and the shorter triangle's perimeter is 30 cm, we can set up the proportion: 30/Perimeter of larger = 3/5. Thus, Perimeter of larger = 30 * (5/3) = 50 cm.
Q7
Medium
If sin(θ) = 0.5 in a right triangle, what is the measure of angle θ?
A.
30 degrees
B.
45 degrees
C.
60 degrees
D.
90 degrees
Show Answer & Explanation
Correct Answer: A
The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. sin(30 degrees) = 0.5, so θ = 30 degrees.
Q8
Hard
In triangle ABC, angle A is 30 degrees, angle B is 60 degrees, and side AC is 10 units long. What is the length of side AB?
A.
5√3
B.
10
C.
5
D.
10√3
Show Answer & Explanation
Correct Answer: A
Using the properties of a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2. Here, AC (10 units) is opposite angle B (60 degrees), so AB (the hypotenuse) is 10 units, and BC (opposite angle A) is 5 units. Thus, AB = 5√3.
Q9
Hard
In triangle ABC, angle A is 30 degrees and angle B is 60 degrees. If the length of side AC is 10 units, what is the length of side AB?
A.
5√3
B.
10
C.
5
D.
10√3
Show Answer & Explanation
Correct Answer: A
Using the properties of 30-60-90 triangles, the side opposite the 30-degree angle (AC) is half the length of the hypotenuse. Thus, AB = AC / sin(30°) = 10 / 0.5 = 20. Therefore, AB = 10√3/2 = 5√3.
Q10
Hard
A right triangle has one angle measuring 30 degrees. If the length of the hypotenuse is 20 units, what is the length of the side opposite the 30-degree angle?
A.
10 units
B.
20 units
C.
5√3 units
D.
15 units
Show Answer & Explanation
Correct Answer: A
In a right triangle, the side opposite a 30-degree angle is half the length of the hypotenuse. Therefore, if the hypotenuse is 20 units, the opposite side is 20/2 = 10 units.
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Similarity Right Triangles and Trigonometry — Common Core (US) Common Core Geometry Practice Questions Online
This page contains 146 practice MCQs for the chapter Similarity Right Triangles and Trigonometry in Common Core (US) Common Core Geometry. The questions are organized by difficulty — 45 easy, 76 medium, 25 hard — so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.