CLEP College Mathematics Practice Test

Explanation

To find the area of a rectangle, we use the formula Area = length * width. In this case, the length is 8 cm and the width is 5 cm. So, the area is 8 * 5 = 40 cm^2. Distractor A results from a sign error, where the student might have subtracted the width from the length instead of multiplying. Distractor C results from a formula misapplication, where the student might have used the formula for the perimeter of a rectangle (2 * length + 2 * width) instead of the area. Distractor D results from an order-of-operations mistake, where the student might have added the length and width before multiplying by a factor.

Explanation

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, the third side must be less than the sum of the other two sides (5 + 7 = 12) and greater than the difference of the other two sides (7 - 5 = 2). This gives a range of 2 to 12. Distractor B applies the concept incorrectly by assuming the third side must be between the lengths of the other two sides. Distractor C results from a sign error, where the difference between the two sides is used as the upper limit instead of the lower limit. Distractor D also applies the concept incorrectly by assuming the third side must be less than the length of the longer side, rather than the sum of the two sides.

Explanation

A rectangle has line symmetry because it can be divided into two equal parts by a vertical or horizontal line. Additionally, it has rotational symmetry of 180 degrees because it looks the same after a 180-degree rotation. Option A is incorrect because it only mentions line symmetry, ignoring the rotational symmetry. Option B is incorrect because a rectangle does not have rotational symmetry of 90 degrees. Option D is incorrect because a rectangle does have symmetry. The correct answer requires understanding of both line and rotational symmetry.

Explanation

The area of a rectangle is calculated by multiplying its length by its width. In this case, the area = length * width = 8 * 5 = 40 square cm. Option A is incorrect because it results from a sign error, where the length and width are added instead of multiplied. Option C is incorrect because it applies the wrong formula, using the sum of the length and width instead of their product. Option D is incorrect because it results from an order-of-operations mistake, where the length is added to the width and then the result is multiplied by 2. The correct formula for the area of a rectangle is applied in option B, which yields the correct answer of 40 square cm.

Explanation

To find the distance between two points (x1, y1) and (x2, y2) in a coordinate plane, we use the distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2). Applying this formula, we get d = √((6 - 2)^2 + (7 - 3)^2) = √(4^2 + 4^2) = √(16 + 16) = √(32) = √(16 * 2) = 4√2 units. Distractor A results from a sign error, as it incorrectly calculates the difference between the x-coordinates. Distractor C results from a formula misapplication, as it incorrectly calculates the distance as the square root of the sum of the squares of the coordinates. Distractor D results from an order-of-operations mistake, as it incorrectly calculates the distance as the square root of the sum of the differences between the coordinates.

Explanation

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, for a triangle with sides of length 5 and 7, the third side must satisfy two conditions: x + 5 > 7 and x + 7 > 5, as well as 5 + 7 > x. Simplifying these inequalities gives x > 2, x > -2, and x < 12. Since length cannot be negative, the condition x > -2 is automatically satisfied for all positive x, and thus the range of possible lengths for the third side is 2 < x < 12. Distractor B results from a sign error, misunderstanding the inequality. Distractor C results from a formula misapplication, only considering one of the inequalities. Distractor D results from an order-of-operations mistake, incorrectly simplifying the inequalities.

Explanation

To find the distance between two points (x1, y1) and (x2, y2), we use the distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2). In this case, d = √((4 - 2)^2 + (6 - 3)^2) = √(2^2 + 3^2) = √(4 + 9) = √13 ≈ 2.83 (rounded to two decimal places) (option B is a sign error, as it incorrectly calculates the distance as the sum of the differences in x and y coordinates, option C is a formula misapplication, as it incorrectly calculates the distance as 2 times the square root of 5, and option D is an order-of-operations mistake, as it incorrectly calculates the distance as the square root of 10).

Explanation

The sum of the interior angles of a triangle is always 180 degrees. Given that angle A is 30 degrees and angle B is 60 degrees, we can calculate angle C as 180 - (30 + 60) = 90 degrees. Distractor A results from a sign error, applying the wrong operation. Distractor C results from a misconception about the properties of triangles, thinking that all angles must be equal. Distractor D results from a formula misapplication, adding the given angles instead of subtracting them from 180 degrees.

Explanation

The figure exhibits line symmetry because one triangle is a reflection of the other across the line of symmetry. This type of symmetry is also known as reflection symmetry. The correct answer, B) Line symmetry across the vertical axis, is the best choice because the line of symmetry is vertical. Distractor A, Rotational symmetry of order 2, is incorrect because rotational symmetry involves rotating the figure by a certain angle, not reflecting it across a line. Distractor C, Rotational symmetry of order 4, is also incorrect because the figure does not exhibit rotational symmetry. Distractor D, Line symmetry across the horizontal axis, is incorrect because the line of symmetry is vertical, not horizontal.

Explanation

The correct answer is A) 180 degrees. This is because the sum of the interior angles of a triangle is always 180 degrees, which is a fundamental property of triangles. This property can be proven by drawing a triangle, extending one of its sides, and observing that the sum of the interior angles is equal to the sum of the angles formed by the extended side and the other two sides. Distractor B) 270 degrees results from a common misconception that the sum of the interior angles of a triangle is equal to the sum of the angles in a quadrilateral. Distractor C) 360 degrees results from a misunderstanding that the sum of the interior angles of a triangle is equal to the number of degrees in a full circle. Distractor D) 90 degrees results from a sign error, where the student incorrectly calculates the sum of the interior angles as being equal to the measure of one right angle.