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Quizzes > High School Quizzes > Mathematics

Unit 3 Similarity & Trigonometry Practice Quiz

Practice key answers for similarity and trigonometry

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting a high school math quiz on similarity and trigonometry concepts.

What does it mean for two triangles to be similar?
They have equal corresponding angles and proportional sides.
They have equal corresponding sides and equal area.
They have equal corresponding sides.
They have congruent corresponding angles but not necessarily proportional sides.
Triangles are similar if their corresponding angles are equal and their sides are proportional. This property ensures that the triangles have the same shape but may differ in size.
Which trigonometric ratio is defined as the ratio of the opposite side to the hypotenuse in a right triangle?
Cosine
Tangent
Secant
Sine
In a right triangle, the sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse. This basic definition is fundamental to trigonometry.
If two triangles are similar with a scale factor of k, how do their corresponding side lengths and perimeters compare?
Side lengths are multiplied by k², and perimeters by k.
Side lengths are multiplied by k, but perimeters remain the same.
Side lengths are multiplied by k, and perimeters by k.
Side lengths remain the same, but perimeters are multiplied by k.
For similar triangles, each side length is scaled by factor k, and since the perimeter is the sum of the side lengths, it is also multiplied by k. Note that the area, however, scales by k², which is not asked here.
Which similarity criterion states that two triangles are similar if they have two pairs of equal angles?
HL (Hypotenuse-Leg) Similarity Criterion
SAS (Side-Angle-Side) Similarity Criterion
SSS (Side-Side-Side) Similarity Criterion
AA (Angle-Angle) Similarity Criterion
The AA (Angle-Angle) criterion is sufficient to establish triangle similarity. When two angles of one triangle are congruent to two angles of another, the triangles are similar because the third angles must also be equal.
What is the approximate value of sin 30°?
1.0
0.5
0.707
0.866
The sine of a 30° angle is a standard trigonometric value, which is 0.5. This value is commonly memorized and applied in many high school math problems involving right triangles.
In two similar triangles, the sides of the smaller triangle are 3, 4, 5. If the largest side of the larger triangle is 15, what is the scale factor?
15
2
5
3
The scaling factor is found by comparing corresponding sides. Since the largest side of the smaller triangle is 5 and the corresponding side in the larger triangle is 15, the scale factor is 15 divided by 5, which equals 3.
A right triangle has an angle of 45°. What is the ratio of the lengths of the opposite side to the adjacent side for that angle?
0.5
2
1
√2
In a 45°-45°-90° triangle, the two legs are congruent, making the ratio of the opposite side to the adjacent side equal to 1. This is a key property of isosceles right triangles.
If sin θ = 0.6 in a right triangle, what is cos θ assuming θ is acute?
1.2
0.4
0.6
0.8
Using the Pythagorean identity, cos²θ = 1 - sin²θ. With sin θ = 0.6, we compute cos²θ = 1 - 0.36 = 0.64, so cos θ = 0.8 for an acute angle.
In a right triangle, if the length of the hypotenuse is 10 and one of the angles is 30°, what is the length of the side opposite the 30° angle?
10
5
7.5
8.66
Since sin 30° equals 0.5, the side opposite the angle is calculated as 0.5 times the hypotenuse. Multiplying 10 by 0.5 gives a length of 5.
For similar triangles, if the ratio of their areas is 9, what is the ratio of their corresponding side lengths?
3
9
81
2
The ratio of the areas of similar triangles is the square of the ratio of their corresponding side lengths. Taking the square root of 9 gives a side length ratio of 3.
In a right triangle, the tangent of an acute angle is 1.2. If the adjacent side is 10, what is the length of the opposite side?
1.2
8.33
10
12
Tangent is the ratio of the opposite side to the adjacent side. Therefore, if tan θ = 1.2 and the adjacent side is 10, the opposite side is 1.2 multiplied by 10, which equals 12.
Which of the following statements about similar triangles is true regarding their areas?
Similar triangles always have the same area.
The ratio of areas is the same as the ratio of corresponding sides.
The ratio of areas equals the cube of the scale factor.
The ratio of areas equals the square of the ratio of corresponding sides.
For similar triangles, the area scales by the square of the side length ratio. This means if the side lengths are scaled by a factor k, the area is scaled by k².
In right triangle trigonometry, which function relates the adjacent side to the hypotenuse?
Cotangent
Sine
Cosine
Tangent
Cosine is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. This fundamental trigonometric function is essential for solving many geometry problems.
A ladder leans against a wall forming a 60° angle with the ground. If the ladder is 10 m long, what is the height it reaches on the wall?
5 m
6.0 m
7 m
8.66 m
Using the sine function, sin 60° is approximately 0.866. Multiplying 10 m by 0.866 gives approximately 8.66 m, which is the vertical height reached on the wall.
In a right triangle with legs of 6 and 8, what is the value of sin for the angle opposite the side of length 6?
0.75
0.6
0.8
0.5
First, determine the hypotenuse using the Pythagorean theorem: √(6² + 8²) = 10. Then, sin of the angle opposite the side of length 6 is 6/10, which equals 0.6.
Two similar right triangles have hypotenuses of lengths 13 and 26 respectively. Which statement best describes the relationship between their corresponding sides?
The triangles do not necessarily have proportional side lengths.
The larger triangle's sides are squared relative to the smaller triangle's sides.
Each side length of the larger triangle is half that of the corresponding side in the smaller triangle.
Each side length of the larger triangle is twice that of the corresponding side in the smaller triangle.
Since the hypotenuse of the larger triangle is 26 and that of the smaller is 13, the scale factor is 26/13 = 2. This means every corresponding side in the larger triangle is twice as long as in the smaller triangle.
Given two similar triangles where the sides of the smaller triangle measure 5, 12, and 13, and the area of the larger triangle is 4 times that of the smaller triangle, what is the scale factor between the triangles?
√2
8
4
2
The area of similar triangles scales by the square of the scale factor. Since the larger triangle's area is 4 times the smaller's, the scale factor is √4, which equals 2.
If in a right triangle the cosine of an angle is 0.3 and the hypotenuse is 20 units, what is the length of the adjacent side?
7.0
0.6
14
6
Cosine is defined as the ratio of the adjacent side to the hypotenuse. Multiplying the hypotenuse (20) by cos θ (0.3) gives an adjacent side length of 6.
A triangle's sides are in the ratio 3:4:5, and it is similar to another triangle with a shortest side of 9 cm. What is the length of the longest side of the larger triangle?
20 cm
12 cm
18 cm
15 cm
With a side ratio of 3:4:5, the smallest side corresponds to 3. Given the smallest side of the larger triangle is 9 cm, the scale factor is 9/3 = 3. Multiplying the largest side (5) by 3 gives 15 cm.
In solving a trigonometry problem involving similar triangles, the sine of an angle was determined to be 0.64. Without a calculator, which of the following is the closest measure for that angle?
50°
60°
30°
40°
The sine of 40° is approximately 0.643, which is very close to 0.64. Recognizing standard trigonometric values helps in estimating angles accurately in such problems.
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Study Outcomes

  1. Apply similarity principles to solve geometric problems involving proportions and scale factors.
  2. Analyze triangle properties to determine unknown side lengths and angles using similarity.
  3. Utilize trigonometric ratios to calculate measures in right triangles.
  4. Interpret diagrams to accurately model and solve similarity and trigonometry problems.
  5. Evaluate solution methods to ensure accuracy and consistency in problem-solving.

Unit 3 Similarity & Trig Answer Key Cheat Sheet

  1. Six Trigonometric Ratios - Meet sine, cosine, tangent and their three cool cousins (cosecant, secant, cotangent) that turn triangles into solvable puzzles. These ratios link angles to side lengths so you can decode any right triangle in a flash. GeeksforGeeks Trigonometry Formulas
  2. Pythagorean Identities - Memorize sin²(θ)+cos²(θ)=1, 1+tan²(θ)=sec²(θ) and 1+cot²(θ)=csc²(θ) to simplify expressions like a math ninja. They're your go‑to shortcuts for rewriting crazy trig terms into neat, solvable forms. GeeksforGeeks Trigonometry Formulas
  3. Angle Sum & Difference Identities - Learn formulas like sin(A±B)=sinA·cosB±cosA·sinB and cos(A±B)=cosA·cosB∓sinA·sinB to tackle sums or differences of angles. These identities let you break big angles into friendlier pieces - and impress your classmates! GeeksforGeeks Trigonometry Formulas
  4. Unit Circle Mastery - Picture a circle of radius one where every angle has a matching point (x,y) that spells out cosine and sine values. This central hub reveals periodic patterns, special angles, and which functions go positive or negative. GeeksforGeeks Trigonometry Formulas
  5. Similar Triangles Concept - When two triangles have equal angles, their sides scale proportionally - like perfectly resized copies of each other. Grasp this idea and watch how it underpins everything from map-making to shadow measurements. The Core Standards - Similarity, Right Triangles & Trigonometry
  6. AA Criterion for Similarity - Use the Angle-Angle rule: if two pairs of corresponding angles match, the triangles are similar. This quick check is a lifesaver for countless geometry and trigonometry problems. The Core Standards - Similarity, Right Triangles & Trigonometry
  7. Law of Sines & Cosines - Go beyond right triangles: the Law of Sines (a/sinA = b/sinB = c/sinC) and Law of Cosines (c² = a²+b²−2ab·cosC) unlock any triangle's missing sides or angles. Perfect for non‑right scenarios! The Core Standards - Similarity, Right Triangles & Trigonometry
  8. Real-World Right Triangle Problems - Apply trig ratios and the Pythagorean theorem to practical puzzles like building ramps, measuring heights, or navigating slopes. Mix in some creativity and suddenly geometry becomes a real adventure. The Core Standards - Similarity, Right Triangles & Trigonometry
  9. Complementary Angle Relationships - Dive into expressions like sin(90°−θ)=cos(θ) and tan(90°−θ)=cot(θ) to flip functions and simplify equations. It's like finding secret passageways in a math maze! The Core Standards - Similarity, Right Triangles & Trigonometry
  10. Dilations & Similarity Transformations - Explore how stretching or shrinking figures from a center point transforms lines and segments while preserving shapes. This concept deepens your grasp of geometric similarity and scaling in the real world. The Core Standards - Similarity, Right Triangles & Trigonometry
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