Question

squares
common core geometry
our final special quadrilateral is perhaps one of the first shapes you ever learned about, the square. although the square seems to be the simplest of all quadrilaterals, it has more properties than all the rest. its definition is as follows.
squares
a square is any quadrilateral with four right angles and four sides of equal length.
exercise #1: based on the definition above, a square is which of the following types of special quadrilaterals. give a reason for why it is or isnt below each:
(a) rectangle (b) rhombus (c) parallelogram (d) trapezoid
squares have many properties and many uses in the real world. lets do a simple application that takes into account many of the properties of squares.
exercise #2: kimi is designing a square garden whose diagonal is exactly 16 feet long. she would like to place fencing around the perimeter of the garden so that it is completely surrounded. if the fencing she likes comes in 12 - foot long rolls, how many rolls will she need and how much fencing will be left over when she is done? show all calculations that lead to your final answer.
exercise #3: in the diagram at the right, square ( abcd ) is drawn and the midpoints of its sides have been marked and connected with segments to form quadrilateral ( efgh ). give an explanation for why ( efgh ) must also be a square.
common core geometry, unit #6 - quadrilaterals - lesson #7
emathinstruction, red hook, ny 12571, © 2018

Explanation:

Step1: Find the side length of the square

Let the side length of the square be \(x\). Using the Pythagorean theorem for a square (\(a^{2}+b^{2}=c^{2}\), where \(a = b=x\) and \(c = 16\) (diagonal)):

$$x^{2}+x^{2}=16^{2}$$

$$2x^{2}=256$$

$$x^{2}=128$$

$$x = 8\sqrt{2}$$

Step2: Calculate the perimeter of the square

The perimeter \(P\) of a square is \(P = 4x\). Substituting \(x = 8\sqrt{2}\):

$$P=4\times8\sqrt{2}=32\sqrt{2}\approx32\times1.414 = 45.248$$

Step3: Determine the number of fencing rolls

Divide the perimeter by the length of each roll (\(12\) feet). \(\frac{45.248}{12}\approx3.77\). Since we can't buy a fraction of a roll, we need \(4\) rolls.

Step4: Calculate the leftover fencing

The total length of \(4\) rolls is \(4\times12 = 48\) feet. Leftover fencing \(L=48 - 45.248 = 2.752\) feet.

Answer:

She will need \(4\) rolls of fencing and there will be approximately \(2.75\) feet of fencing left over.