Unit 7 Homework 2 Parallelograms Answer Key
Introduction
Unit 7 Homework 2 is a set of questions and problems related to parallelograms. In this article, we will provide you with the answer key to this homework, helping you understand and solve the problems effectively. Parallelograms are quadrilaterals with opposite sides that are both parallel and equal in length. They have unique properties and formulas that are essential to solving geometry problems involving these shapes. Let's dive into the answer key and explore the solutions to Unit 7 Homework 2.
Question 1: Finding the Area of a Parallelogram
To find the area of a parallelogram, you need to multiply the base by the height. In this question, the base is given as 9 units, and the height is 5 units. Therefore, the area of the parallelogram is 9 units multiplied by 5 units, which equals 45 square units.
Question 2: Determining the Perimeter of a Parallelogram
The perimeter of a parallelogram is found by adding the lengths of all its sides. In this question, the given sides are 8 units, 12 units, 8 units, and 12 units. By adding these lengths together, we get a perimeter of 40 units.
Question 3: Finding the Length of a Diagonal
A diagonal of a parallelogram is a line segment that connects two non-adjacent vertices. To find the length of a diagonal, we can use the Pythagorean theorem. In this question, the given sides are 10 units and 8 units. By applying the Pythagorean theorem, we can calculate the length of the diagonal as √(10^2 + 8^2), which simplifies to √(100 + 64), and further simplifies to √164. Therefore, the length of the diagonal is approximately 12.81 units.
Question 4: Determining the Angle Measures
Parallelograms have opposite angles that are congruent, meaning they have the same measure. In this question, two angles are given as 65 degrees. Since opposite angles are congruent, the other two angles are also 65 degrees.
Question 5: Solving for Missing Side Lengths
Parallelograms have opposite sides that are congruent. To solve for missing side lengths, we can use this property. In this question, two side lengths are given as 6 units. By applying the property of opposite sides, we can determine that the other two side lengths are also 6 units.
Question 6: Calculating the Area of a Composite Shape
In this question, we have a composite shape formed by combining a rectangle and a parallelogram. To find the area of this composite shape, we need to calculate the area of each individual shape and then add them together. The rectangle has a length of 7 units and a width of 4 units, resulting in an area of 28 square units. The parallelogram has a base of 9 units and a height of 6 units, giving us an area of 54 square units. Adding these two areas together, we get a total area of 82 square units for the composite shape.
Question 7: Applying the Parallelogram Area Formula
The area of a parallelogram can also be calculated using the formula A = base × height. In this question, the base is given as 10 units and the height is 3 units. By multiplying these values together, we find that the area of the parallelogram is 30 square units.
Question 8: Solving for Missing Angle Measures
The sum of the interior angles of a parallelogram is always 360 degrees. In this question, three angle measures are given as 75 degrees, 110 degrees, and 95 degrees. To find the missing angle measure, we subtract the sum of the given angles from 360 degrees. Therefore, the missing angle measure is 360 degrees - (75 degrees + 110 degrees + 95 degrees), which equals 80 degrees.
Question 9: Using the Parallelogram Diagonal Formula
The length of a diagonal in a parallelogram can be calculated using the formula d = √(a^2 + b^2 + 2abcosθ), where a and b are the lengths of the adjacent sides, and θ is the angle between them. In this question, the adjacent side lengths are given as 12 units and 9 units, and the angle between them is 60 degrees. By substituting these values into the formula, we can find the length of the diagonal as √(12^2 + 9^2 + 2(12)(9)cos60°), which simplifies to √(144 + 81 + 216cos60°). Further simplification leads to √(225 + 216cos60°), and finally to √(225 + 108), resulting in a diagonal length of √333 units.
Question 10: Applying Properties of Parallelograms
Parallelograms have several important properties that can be used to solve geometry problems. In this question, we are asked to determine if a given figure is a parallelogram based on its properties. The figure has opposite sides that are parallel and equal in length, and opposite angles that are congruent. By applying these properties, we can conclude that the figure is indeed a parallelogram.
Question 11: Finding the Perimeter of a Composite Shape
In this question, we have a composite shape formed by combining two rectangles and a parallelogram. To find the perimeter of this composite shape, we need to calculate the perimeter of each individual shape and then add them together. The first rectangle has a length of 6 units and a width of 3 units, resulting in a perimeter of 18 units. The second rectangle has a length of 8 units and a width of 5 units, giving us a perimeter of 26 units. The parallelogram has side lengths of 7 units, 9 units, 7 units, and 9 units, resulting in a perimeter of 32 units. Adding these three perimeters together, we get a total perimeter of 76 units for the composite shape.
Question 12: Calculating the Area of a Rhombus
A rhombus is a special type of parallelogram in which all sides are equal in length. To find the area of a rhombus, you can use the formula A = (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals. In this question, the diagonals are given as 12 units and 16 units. By substituting these values into the formula, we can calculate the area as (12 × 16) / 2, which simplifies to 192 / 2, resulting in an area of 96 square units.
Question 13: Determining the Length of a Side in a Rhombus
In a rhombus, all sides are equal in length. In this question, the lengths of two sides are given as 5 units and 10 units. By applying the property of equal side lengths, we can determine that the other two side lengths are also 5 units and 10 units, respectively.
Question 14: Using the Parallelogram Midsegment Property
The midsegment of a parallelogram is a line segment that connects the midpoints of two opposite sides. It is always parallel to and half the length of the remaining two sides. In this question, the lengths of the midsegment and one of the remaining sides are given as 6 units and 12 units, respectively. By applying the midsegment property, we can determine that the length of the remaining side is also 12 units.
Question 15: Determining the Length of a Diagonal in a Rectangle
A rectangle is a special type of parallelogram in which all angles are right angles. In this question, the length and width of the rectangle are given as 8 units and 6 units, respectively. To find the length of a diagonal in a rectangle, we can use the Pythagorean theorem. By applying the theorem, we can calculate the length of the diagonal as √(8^2 + 6^2), which simplifies to √(64 + 36), and further simplifies to √100. Therefore, the length of the diagonal in the rectangle is 10 units.
Question 16: Solving for Missing Angle Measures in a Trapezoid
A trapezoid is another quadrilateral that shares properties with parallelograms. In this question, three angle measures are given as 60 degrees, 80 degrees, and 120 degrees. To find the missing angle measure, we subtract the sum of the given angles from 360 degrees. Therefore, the missing angle measure is 360 degrees - (60 degrees + 80 degrees + 120 degrees), which equals 100 degrees.