a Calculate the lengths of the sides using the Ruler Postulate.
B
b The quadrilateral and the square together create triangles in the corners. They can be of help.
C
c Calculate the length of one of the sides using the Distance Formula.
A
a P=16 units and A=16 square units.
B
b Yes, see solution.
C
c A=8 square units and P≈ 11.31 units. The smaller square's area is half of the area of the bigger square.
Practice makes perfect
a Let's begin by drawing the square in a diagram.
Since we know that it is a square, we only need to find the length of one of the sides to be able to calculate the perimeter and the area. Let's find the length of the side LM. The points L and M have the same y-coordinate therefore the segment is horizontal. We can calculate its length using the Ruler Postulate.
We now know that the length of the square's sides is 4 units. Let's call the side s_b. The square's area we calculate using the formula A=s^2. For this square the area is
A=s_b^2=4^2=16 square units.
The perimeter is the sum of lengths of the square's sides. The square has 4 sides each with the length 4 units and consequently its perimeter is
P=4* s_b=4 * 4 = 16 units.
b Let's now draw lines between the midpoints of the sides of the square and thus create a quadrilateral. We name the midpoints A, B, C, and D.
We now want to know the coordinates for corners of the quadrilateral. Since each side of the square is 4 units, the midpoints are 2 units away. Let's draw the quadrilateral without the square around it.
We want to know is this quadrilateral is a square. To check this, we will need to check that all four sides have the same lengths and that each angle is a right angle. To check the side lengths, we will need to use the Distance Formula. Let's start by finding the length of AB.
Similarly, we use the Distance Formula to find the other three lengths:
Segment
AB
BC
CD
DA
Length
2sqrt(2)
2sqrt(2)
2sqrt(2)
2sqrt(2)
Since all four sides in the quadrilateral have the same length, we now want to know if it has four right angles. We have a right angle if two sides are perpendicular and the product of their slopes is - 1. Let's calculate the slope of the side DA. For this we use the Slope Formula.
We can continue to use the Slope Formula to find the slopes of the other sides of the quadrilateral as well.
Segment
AB
BC
CD
DA
Slope
-1
1
-1
1
When we multiply the two slopes, m_(AB) and m_(DA), with each other we get
m_(AB)* m_(DA)=1* (- 1)=- 1.
Since the product is - 1 we know that the lines are perpendicular and that the angle is right. Now we check the others:
Segments
AB & BC
BC & CD
CD & DA
DA & AB
Product of Slopes
-1
-1
-1
-1
Since all four products are -1, we have four right angles. A quadrilateral with four sides of equal length and with four right angles is called a square.
c We know that we have a square. Let's calculate the length of its side, s_s. We calculate the length of the side AB and we do that using the Distance Formula.
The length of the square's sides is s_s=sqrt(8) units. We calculate the area of a square using the formula A=s^2. Here the area is
A=s_s^2=( sqrt(8) )^2=8 square units.
We find the perimeter by adding the sides of the square. With four sides each having the length sqrt(8) units the perimeter is
P=4* s_s=4* sqrt(8)≈ 11.31 units.
The square we started out with had the area 16 square units. This smaller square's area is 8 which is half the area of the bigger square.
