The following is a list of triangles: I. Right triangles, II. Isosceles triangles, III. Equilateral triangles. A pair of triangles from which of these groups must be similar to each other?
- A. I only
- B. II only
- C. III only
- D. I and III only
Correct Answer & Rationale
Correct Answer: C
Triangles from group III, equilateral triangles, are always similar to each other because they all have equal angles of 60 degrees, regardless of their size. Group I, right triangles, can vary significantly in angle measures beyond the right angle, so not all right triangles are similar. Similarly, group II, isosceles triangles, can have different base angles, leading to non-similar triangles. Thus, while right and isosceles triangles can share properties, only equilateral triangles guarantee similarity across the group. Therefore, option C accurately identifies the group with universally similar triangles.
Triangles from group III, equilateral triangles, are always similar to each other because they all have equal angles of 60 degrees, regardless of their size. Group I, right triangles, can vary significantly in angle measures beyond the right angle, so not all right triangles are similar. Similarly, group II, isosceles triangles, can have different base angles, leading to non-similar triangles. Thus, while right and isosceles triangles can share properties, only equilateral triangles guarantee similarity across the group. Therefore, option C accurately identifies the group with universally similar triangles.
Other Related Questions
Josh takes 6 hours to paint a room. Margaret can paint the same room in 4 hours. Assuming their individual rates do not change, how long will it take them to paint the room together?
- A. 1.5 hours
- B. 2.4 hours
- C. 4.8 hours
- D. 5 hours
- E. 10 hours
Correct Answer & Rationale
Correct Answer: B
To determine how long it takes Josh and Margaret to paint the room together, we first calculate their individual rates. Josh paints at a rate of \( \frac{1}{6} \) of the room per hour, while Margaret paints at \( \frac{1}{4} \) of the room per hour. Combined, their rates are: \[ \frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \] This means together they paint \( \frac{5}{12} \) of the room per hour. To find the time taken to complete one room, we take the reciprocal of their combined rate: \[ \text{Time} = \frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4 \text{ hours} \] Option A (1.5 hours) is too short, as it implies a higher combined rate than possible. Option C (4.8 hours) suggests they are slower than working alone, which is incorrect. Option D (5 hours) is also longer than their combined effort should take, and Option E (10 hours) is excessively long, indicating a misunderstanding of their rates. Thus, 2.4 hours accurately reflects their collaborative efficiency.
To determine how long it takes Josh and Margaret to paint the room together, we first calculate their individual rates. Josh paints at a rate of \( \frac{1}{6} \) of the room per hour, while Margaret paints at \( \frac{1}{4} \) of the room per hour. Combined, their rates are: \[ \frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \] This means together they paint \( \frac{5}{12} \) of the room per hour. To find the time taken to complete one room, we take the reciprocal of their combined rate: \[ \text{Time} = \frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4 \text{ hours} \] Option A (1.5 hours) is too short, as it implies a higher combined rate than possible. Option C (4.8 hours) suggests they are slower than working alone, which is incorrect. Option D (5 hours) is also longer than their combined effort should take, and Option E (10 hours) is excessively long, indicating a misunderstanding of their rates. Thus, 2.4 hours accurately reflects their collaborative efficiency.
Through which pair of points could a line of best fit be drawn for the data on the scatterplot?
- A. (0, 36) and (11, 74)
- B. (1, 39) and (6, 60)
- C. (5, 50) and (6, 60)
- D. (6, 60) and (8, 60)
- E. (8, 60) and (11, 74)
Correct Answer & Rationale
Correct Answer: A
Option A, with points (0, 36) and (11, 74), shows a significant range in both x and y values, indicating a strong upward trend that aligns well with the overall direction of the data. Option B, while showing an upward trend, has a narrower range and may not represent the overall data as effectively. Option C features two points that are too close together, limiting their ability to define a clear line of best fit. Option D includes points with the same y-value, suggesting a horizontal line that does not capture the data's trend. Option E, like A, has a valid upward trend but does not span the data range as effectively as A.
Option A, with points (0, 36) and (11, 74), shows a significant range in both x and y values, indicating a strong upward trend that aligns well with the overall direction of the data. Option B, while showing an upward trend, has a narrower range and may not represent the overall data as effectively. Option C features two points that are too close together, limiting their ability to define a clear line of best fit. Option D includes points with the same y-value, suggesting a horizontal line that does not capture the data's trend. Option E, like A, has a valid upward trend but does not span the data range as effectively as A.
In a survey of 300 people who were randomly sampled from a well-defined population, 60 said that they read a newspaper daily. If 1,000 people had been randomly sampled from the same population and asked the same question, how many would be expected to say they read a newspaper daily?
- A. 180
- B. 200
- C. 360
- D. 600
- E. 760
Correct Answer & Rationale
Correct Answer: A
To determine how many people would be expected to read a newspaper daily in a larger sample, we first find the proportion from the initial survey. Out of 300 people, 60 read a newspaper daily, resulting in a proportion of 60/300 = 0.2 or 20%. Applying this proportion to a sample of 1,000 people, we calculate 20% of 1,000, which is 200. Therefore, option B (200) is the expected number. Other options are incorrect as follows: - A (180) underestimates the proportion. - C (360) overestimates, assuming a higher reading rate. - D (600) and E (760) are significantly higher, suggesting an unrealistic increase in readership.
To determine how many people would be expected to read a newspaper daily in a larger sample, we first find the proportion from the initial survey. Out of 300 people, 60 read a newspaper daily, resulting in a proportion of 60/300 = 0.2 or 20%. Applying this proportion to a sample of 1,000 people, we calculate 20% of 1,000, which is 200. Therefore, option B (200) is the expected number. Other options are incorrect as follows: - A (180) underestimates the proportion. - C (360) overestimates, assuming a higher reading rate. - D (600) and E (760) are significantly higher, suggesting an unrealistic increase in readership.
What is the sum of the two polynomials? 4x² + 3x + 5 + x² + 6x - 3?
- A. 4x² + 9x + 2
- B. 5x² + 9x + 2
- C. 5x² + 9x + 8
- D. 4x² + 9x² + 2
- E. 5x² + 9x² + 8
Correct Answer & Rationale
Correct Answer: B
To find the sum of the polynomials \(4x^2 + 3x + 5\) and \(x^2 + 6x - 3\), we combine like terms. 1. For \(x^2\) terms: \(4x^2 + x^2 = 5x^2\). 2. For \(x\) terms: \(3x + 6x = 9x\). 3. For constant terms: \(5 - 3 = 2\). Thus, the resulting polynomial is \(5x^2 + 9x + 2\), which corresponds to option B. Option A incorrectly adds the \(x^2\) terms, leading to an incorrect polynomial. Option C miscalculates the constant term. Option D mistakenly adds the \(x^2\) terms incorrectly and does not follow proper polynomial addition. Option E also miscalculates by incorrectly summing the \(x^2\) terms and the constants.
To find the sum of the polynomials \(4x^2 + 3x + 5\) and \(x^2 + 6x - 3\), we combine like terms. 1. For \(x^2\) terms: \(4x^2 + x^2 = 5x^2\). 2. For \(x\) terms: \(3x + 6x = 9x\). 3. For constant terms: \(5 - 3 = 2\). Thus, the resulting polynomial is \(5x^2 + 9x + 2\), which corresponds to option B. Option A incorrectly adds the \(x^2\) terms, leading to an incorrect polynomial. Option C miscalculates the constant term. Option D mistakenly adds the \(x^2\) terms incorrectly and does not follow proper polynomial addition. Option E also miscalculates by incorrectly summing the \(x^2\) terms and the constants.