Quadrilateral GHIJ is a scale drawing of quadrilateral ABCD with a scale factor r. Describe the following statements as always true, sometimes true, or never true, and justify your answer.
a. AB/GH = BC/HI
b. perimeter (GHIJ) = r * perimeter (ABCD)
c. Area (GHIJ) = r * area (ABCD)
d. r < 0
a true
b true
c false
d false (unless you consider r<0 as a reflection as well as a scaling. Then it could be true sometimes)
what is then of quadrilateral GHIJ
a. Sometimes true. The ratio AB/GH is equal to the ratio BC/HI if and only if all corresponding sides are proportional in the two quadrilaterals. If the scale factor r is the same for all sides, then the statement is always true. However, if the sides are not proportional or if the scale factor is different for different sides, then the statement is sometimes true.
b. Always true. The perimeter of a shape is simply the sum of its side lengths. In a scale drawing, all corresponding sides are multiplied by the same scale factor. Therefore, if the perimeter of ABCD is P, then the perimeter of GHIJ (a scale drawing with scale factor r) will be r * P, regardless of the specific shape or scale factor.
c. Sometimes true. The area of a quadrilateral is determined by the lengths of its sides and the included angles. If the scale factor r applies to all sides and angles, then the statement is always true. However, if the shape or scale factor causes a change in angles or if the scale factor is different for different sides, then the statement is sometimes true.
d. Never true. The scale factor r represents the enlargement or reduction of the scale drawing compared to the original shape. It is always a positive value or zero, since negative values would imply a reflection of the shape rather than a simple scaling. Therefore, r is always greater than or equal to 0 and is never negative.
a. Sometimes true.
This statement is sometimes true because if quadrilateral ABCD and GHIJ are both similar, meaning their corresponding angles are equal and their corresponding sides are proportional, then the ratio AB/GH would indeed be equal to the ratio BC/HI. However, if the quadrilaterals are not similar, then the ratios would not be equal.
To determine whether or not this statement is true in a specific case, you would need to compare the corresponding sides AB and GH, as well as BC and HI. If the ratios of these sides are equal, then the statement is true. Otherwise, it is false.
b. Always true.
This statement is always true because when you scale a figure by a factor of r, all the sides are multiplied by r. Therefore, the perimeter of the scaled figure, GHIJ, will be r times the perimeter of the original figure, ABCD.
To calculate the perimeters, you would simply add up the lengths of all the sides in each figure and then multiply the perimeter of ABCD by r to get the perimeter of GHIJ.
c. Always true.
This statement is always true because when you scale a figure by a factor of r, the area of the scaled figure is multiplied by r^2. Therefore, the area of the scaled figure, GHIJ, will be r times the area of the original figure, ABCD.
To calculate the areas, you would need to find the areas of both ABCD and GHIJ using appropriate formulas (depending on the shape of the quadrilaterals) and then multiply the area of ABCD by r to get the area of GHIJ.
d. Never true.
The scale factor, represented by r, is a positive number. The scale factor describes how much larger or smaller the scaled figure is compared to the original figure. It cannot be negative because it represents a proportional change in size.
Therefore, r is always greater than zero (r > 0) and cannot be negative (r < 0). So, the statement "r < 0" is never true.