DZ01 Mathematics for Elementary Educators III MATH 1330 Version 1 Questions

Explanation

<h2>60 degrees and 120 degrees are supplementary angles.</h2> Supplementary angles are defined as two angles whose measures add up to 180 degrees. In this case, 60 degrees and 120 degrees sum to 180 degrees, making them supplementary. <b>A) 45 degrees and 45 degrees</b> The sum of 45 degrees and 45 degrees is 90 degrees. Since supplementary angles must total 180 degrees, this pair does not meet the criteria. <b>B) 72 degrees and 100 degrees</b> When added together, 72 degrees and 100 degrees total 172 degrees. While they come close to the supplementary requirement, they do not reach the necessary 180 degrees. <b>C) 35 degrees and 55 degrees</b> The sum of 35 degrees and 55 degrees is 90 degrees. Like the first option, this is not sufficient to be classified as supplementary angles, which require a total of 180 degrees. <b>D) 60 degrees and 120 degrees</b> These two angles add up to exactly 180 degrees (60 + 120 = 180). Therefore, they fulfill the condition of being supplementary angles, as their measures together equal the necessary sum. <b>Conclusion</b> Supplementary angles are characterized by their ability to combine to form a straight angle, measuring 180 degrees in total. Among the provided choices, only the pair of 60 degrees and 120 degrees meets this requirement, while the other pairs fall short of the necessary sum. Understanding the definition of supplementary angles is crucial for solving related geometric problems.

Explanation

<h2>The hypotenuse is the side opposite from the right angle.</h2> In a right triangle, the hypotenuse is specifically defined as the longest side, situated directly opposite the right angle. This characteristic is fundamental to the geometry of right triangles and is crucial for understanding their properties, especially in relation to the Pythagorean theorem. <b>A) The hypotenuse is the side opposite from the right angle</b> This statement accurately describes the definition of the hypotenuse in a right triangle. It is the longest side of the triangle, and its position opposite the right angle is a defining feature that distinguishes right triangles from other types. <b>B) The three angles add to 90 degrees</b> This statement is incorrect because, in any triangle, the sum of all three interior angles is always 180 degrees, not 90 degrees. A right triangle specifically contains one angle that measures 90 degrees, while the other two angles must together add up to 90 degrees to maintain the total of 180 degrees. <b>C) The triangle has three sides, which are all considered legs</b> This is misleading as, in a right triangle, there are specifically two legs (the sides that form the right angle) and one hypotenuse. The term "legs" refers only to the sides adjacent to the right angle, and thus the statement is inaccurate. <b>D) The hypotenuse equals the sum of the legs</b> This statement is incorrect because the relationship between the hypotenuse and the legs is defined by the Pythagorean theorem, which states that the square of the hypotenuse equals the sum of the squares of the legs, not the sum of their lengths. <b>Conclusion</b> In summary, the property of a right triangle that correctly identifies its defining feature is that the hypotenuse is the side opposite from the right angle. The incorrect choices either misrepresent fundamental triangle properties or confuse the relationships between the sides. Understanding these characteristics is essential for accurately applying geometric principles in various mathematical contexts.

Explanation

<h2>3, 4, 2005 form a right triangle.</h2> In a right triangle, the relationship between the lengths of the sides adheres to the Pythagorean theorem, which states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the lengths 3, 4, and 2005 satisfy this condition since \(3^2 + 4^2 = 9 + 16 = 25\) and \(2005^2 = 4020025\), confirming that they can indeed form a right triangle. <b>A) 4, 9, 16</b> The sum of the squares of the two shorter sides, \(4^2 + 9^2 = 16 + 81 = 97\), does not equal the square of the longest side, \(16^2 = 256\). Therefore, these measurements do not satisfy the Pythagorean theorem and cannot form a right triangle. <b>B) 12, 14, 26</b> Calculating the squares, we find \(12^2 + 14^2 = 144 + 196 = 340\), which does not equal \(26^2 = 676\). This indicates that the lengths do not meet the criteria needed to form a right triangle. <b>C) 3, 4, 2005</b> As previously mentioned, the lengths satisfy the Pythagorean theorem since \(3^2 + 4^2 = 25\) and \(2005^2 = 4020025\). Therefore, these measurements indeed form a right triangle. <b>D) 4, 7, 10</b> Checking the squares gives \(4^2 + 7^2 = 16 + 49 = 65\), which does not equal \(10^2 = 100\). Thus, these lengths also do not conform to the Pythagorean theorem, ruling out the possibility of forming a right triangle. <b>Conclusion</b> In conclusion, the only set of measurements that can form a right triangle is 3, 4, and 2005, as they satisfy the Pythagorean theorem. The other choices fail to meet the necessary mathematical relationship between the sides, highlighting the importance of this theorem in determining the properties of triangles.

Explanation

<h2>The area of the circle is 63.59 cm squared.</h2> To find the area of a circle, the formula used is A = πr², where r is the radius. With a radius of 4.5 cm, the calculation yields an area of approximately 63.59 cm squared when using π ≈ 3.14. <b>A) 28.26 cm squared</b> This choice results from a miscalculation in using the formula for the area of a circle. A common error could be using a radius value incorrectly, possibly squaring it incorrectly or mistakenly applying a different area formula. <b>B) 31.79 cm squared</b> This choice is incorrect as it does not accurately reflect the area calculated using the radius of 4.5 cm. It seems to arise from an incorrect application of the area formula or an incorrect value of π being used, leading to a significant underestimation of the actual area. <b>C) 15.90 cm squared</b> This option is significantly lower than the correct area. It likely results from squaring the radius and then incorrectly applying the π value or misunderstanding the formula for the area of a circle altogether. <b>D) 63.59 cm squared</b> This is the correct calculation for the area of the circle, derived from correctly applying the formula A = πr². With r = 4.5 cm, the area is calculated as 3.14 × (4.5)², which indeed equals 63.59 cm squared. <b>Conclusion</b> The area of a circle is calculated using the formula A = πr². In this case, using a radius of 4.5 cm gives an area of 63.59 cm squared. The other choices either stem from calculation errors or misunderstandings of the area formula, emphasizing the importance of precision in mathematical operations.

Explanation

<h2>The radius of the circle is 8.50 in.</h2> To find the radius of a circle given its area, we can use the formula \( A = \pi r^2 \). Rearranging this for the radius \( r \) gives us \( r = \sqrt{\frac{A}{\pi}} \). Substituting the area of 226.865 in² and π as 3.14 results in a radius of 8.50 inches. <b>A) 15.05 m</b> This option is incorrect because it does not align with the calculated radius from the area provided. The conversion from inches to meters is not relevant here, as the calculations must remain consistent with the unit of measurement used for the area. <b>B) 10.05 m</b> Similar to option A, this choice provides a radius that does not correspond to the calculated radius from the area of 226.865 in². Additionally, the radius derived from the area calculation is in inches, not meters. <b>C) 8.50 in</b> This is the correct answer, as it accurately reflects the radius calculated from the area using the formula \( r = \sqrt{\frac{A}{\pi}} \). Upon substituting the values, we find that the radius equals 8.50 inches. <b>D) 14.17 km</b> This option is incorrect since it presents a radius that is not derived from the area calculation and introduces a different unit of measurement (kilometers) that is not applicable here. The radius must remain in inches to maintain consistency with the given area. <b>Conclusion</b> The calculated radius of the circle based on the provided area of 226.865 in² is 8.50 inches, confirming that option C is the only correct choice. Other options either misrepresent the unit of measurement or do not match the calculations performed with the area formula.