Question

Rectangle WXYZ has a perimeter of 16 units and an area of 15 square units. Part A- Graph WXYZ on a sheet of graph paper. Write a composition of rigid motions describing two reflections of WXYZ across parallel lines of your choosing. Graph and label the parallel lines W'X'Y'Z' and W"X"Y"Z". Part B- Write a single rigid motion that is equivalent to the composition of rigid motions in Part A. Justify your answer. Part C- Compare the perimeter and area of WXYZ and W"X"Y"Z". What can you conclude about the effect of translation on the properties of figures?

Solution

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Step 1

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We solve for the dimensions, ll and ww, of the rectangle first. We are given that the perimeter is 16 units so we can write:

2l+2w=162l+2w=16

l+w=8(1)\color{#4257b2}{l+w=8}\tag{1}

We are also given that the area is 15 square units so we can write:

lw=15(2)\color{#4257b2}{ lw=15}\tag{2}

Solve for ww using (1):

w=8lw=8-l

Substitute to (2) and solve for ll:

l(8l)=15l(8-l)=15

8ll2=158l-l^2=15

0=l28l+150= l^2-8l+15

0=(l5)(l3)0=(l-5)(l-3)

l=3,5l=3,5

Solve for the corresponding widths:

w=83=5 and w=85=3w=8-3=5\text{ and }w=8-5=3

Using appropriate measures, l=5l=5 units and w=3w=3 units.

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