Question

unit 1 lesson 11 practice problems pg. 59
problem 1
if two rectangles have the same perimeter, do they have to be congruent? explain how you know.
problem 2
draw two rectangles that have the same area, but are not congruent.
problem 3
for each pair of shapes, decide whether or not it appears that the two shapes are congruent. explain your reasoning.
a.
b.

Explanation:

Problem 1

Step1: Recall rectangle perimeter formula

The perimeter of a rectangle is $P = 2(l + w)$ where $l$ is length and $w$ is width.

Step2: Provide counter - example

A rectangle with $l = 1$ and $w = 5$ has $P=2(1 + 5)=12$. A rectangle with $l = 2$ and $w = 4$ has $P = 2(2+4)=12$. But they are not congruent as side - lengths differ. So, no, two rectangles with the same perimeter don't have to be congruent.

Problem 2

A rectangle with $l = 2$ and $w = 4$ has area $A=2\times4 = 8$. A rectangle with $l = 1$ and $w = 8$ has area $A=1\times8 = 8$. You can draw a rectangle with length 2 units and width 4 units and another with length 8 units and width 1 unit. They have the same area but are not congruent.

Problem 3

a.

Step1: Recall congruence definition

Congruent shapes have same size and shape. These two ellipses may have different sizes (major and minor axes can vary), so they are not necessarily congruent. We cannot be sure they are congruent just by looking as their dimensions (lengths of axes) are not given.

b.

Step1: Analyze the polygons

By looking at the two polygons, if we assume they are drawn to scale and we can match up the corresponding sides and angles. Since the note on the right - hand side says "They are", we can assume that through rigid motions (translations, rotations, reflections) one can be mapped onto the other, so they are congruent.

Answer:

Problem 1

No. Counter - example: rectangle with sides 1, 5 and rectangle with sides 2, 4 have same perimeter but are not congruent.

Problem 2

Draw a rectangle with length 2 and width 4 and another with length 8 and width 1.

Problem 3

a.

Not necessarily congruent as dimensions are not given.

b.

Congruent as one can be mapped onto the other by rigid motions.