Question

3.1.3 geometry homework
name:
3 - 29. rashna is puzzled. she is working with the parallelogram drawn at right and wants to make
it smaller instead of bigger.
a what should she do if she wants the sides of her new figure to be half as long as the
original sides? what zoom factor should she use? find the dimensions of her new figure
16
11
b while drawing some other shapes, rashna ended up with a shape congruent to the original
parallelogram. what is the common ratio between pairs of corresponding sides?
3 - 30. enlarge the shape below using a zoom factor of 2. give the lengths of the enlarged figure. then find the
perimeter and area of both shapes. what do you notice when you compare the perimeters? the areas?
perimeter of original:
________
original
enlarged
area of original:
________
perimeter of enlarged:
________
area of enlarged:
________
3 - 31. solve each equation below. show all work and check your answer.
a \\( \frac { 14 } { 9 } = \frac { 4 } { y } \\)
b \\( \frac { 10 } { m } = \frac { 5 } { 11 } \\)
c. \\( \frac { t - 2 } { 12 } = \frac { 7 } { 8 } \\)
d \\( \frac { x + 1 } { 2 } = \frac { x } { 8 } \\)

Explanation:

Step1: Calculate the perimeter of the original trapezoid

The formula for the perimeter of a trapezoid is \(P = a + b + c + d\). For the original trapezoid with sides \(4\), \(6\), \(5\), and \(9\), we have \(P_{original}=4 + 6+5 + 9=24\).

Step2: Calculate the area of the original trapezoid

The formula for the area of a trapezoid is \(A=\frac{(a + b)h}{2}\), where \(a = 6\), \(b = 9\), and \(h = 4\). So \(A_{original}=\frac{(6 + 9)\times4}{2}=\frac{15\times4}{2}=30\).

Step3: Calculate the side - lengths of the enlarged trapezoid

Using a zoom factor of \(2\), the new side - lengths are \(4\times2 = 8\), \(6\times2=12\), \(5\times2 = 10\), and \(9\times2=18\).

Step4: Calculate the perimeter of the enlarged trapezoid

Using the perimeter formula \(P = a + b + c + d\) with the new side - lengths, \(P_{enlarged}=8 + 12+10 + 18 = 48\).

Step5: Calculate the area of the enlarged trapezoid

Using the area formula \(A=\frac{(a + b)h}{2}\) with \(a = 12\), \(b = 18\), and \(h = 8\), \(A_{enlarged}=\frac{(12 + 18)\times8}{2}=\frac{30\times8}{2}=120\).

Answer:

Perimeter of Original: \(24\)
Area of Original: \(30\)
Perimeter of Enlarged: \(48\)
Area of Enlarged: \(120\)

When comparing the perimeters: The perimeter of the enlarged figure is \(2\) times the perimeter of the original figure (\(\frac{48}{24}=2\)).
When comparing the areas: The area of the enlarged figure is \(4\) times the area of the original figure (\(\frac{120}{30} = 4\)).