Answer:
A square is defined as a quadrilateral with 4 sides of the same length and 4 right angles. To verify the lengths of the sides, you can use the distance formula. To find the angles, you can use the slopes of the line segments and make sure sides that form an angle have opposite reciprocal slopes.
Step-by-step explanation:
First, find the coordinates of each point. You're provided with the scale and you can just count, but to make it easier, you can label the axes.
A: (0,8)
B: (6,6)
C: (4,0)
D: (-2,2)
The distance formula is used to find the distance between any two points on the coordinate form. Here it is:
d=sqrt((y2-y1)^2 + (x2-x1)^2)
d=distance, and (x1, y1) and (x2, y2) are the points.
Find the length of each side. I'll demonstrate with two sides, AB and BC.
AB: (x1,y1)=(0,8) and (x2,y2)=(6,6)
d=sqrt((6-8)^2 + (6-0)^2)
=sqrt((-2)^2 + 6^2)
=sqrt(4+36)
=sqrt(40)=2sqrt10
BC: (x1,y1)=(6,6) and (x2,y2)=(4,0)
d=sqrt((4-6)^2 + (0-6)^2)
=sqrt((-2)^2 + (-6)^2)
=sqrt(4+36)
=sqrt(40)=2sqrt10
Do the same thing with sides CD and AD, you'll see that they both also have a length of 2sqrt10 inches. (You don't really have to simplify, I just like to.)
To verify if the sides are perpendicular, you need to find the slopes of the sides and make sure that the product of the slopes of each pair of sides that form a vertex is -1. In other words, each pair of sides that form a vertex have to have opposite reciprocal slopes. I'll demonstrate with 2 vertecies, A and B.
To find the slope, you have to use the formula m=(y2-y1)/(x2-x1). m is the slope, and again, (x1, y1) and (x2, y2) are the points you're plugging in.
To find the measure of angle A, you need to find the slopes of sides AD and AB. (These are the sides that meet to form the angle.)
AD: (x1,y1)=(0,8) and (x2,y2)=(-2,2)
m = (2 - 8)/(-2 - 0)
= -6/-2
= 3
AB: (x1,y1)=(0,8) and (x2,y2)=(6,6)
m = (6 - 8)/(6 - 0)
= -2/6
= -1/3
So AD has a slope of 3 and AB has a slope of -1/3. Now you need to make sure they have a product of -1.
-1/3 * 3 = -1
So yes, it works! As I mentioned before, you can also think of them as opposite reciprocals.
The reciprocal of 3 is 1/3. The opposite of 1/3 is -1/3. So yes, the sides are perpendicular.
Let's do it again with angle B. We have to find the slope of BC now, since we already know the slope of AB.
BC: (x1,y1)=(6,6) and (x2,y2)=(4,0)
m = (0 - 6)/(4-6)
= -6/-2
=3
So the slope is exactly the same as the slope of AD. A square is a parallelogram, so it has two pairs of opposite parallel sides. We already verified that 3 and -1/3 are opposite reciprocals, so we now know that angle B has a measure of 90 degrees as well. You can do the same for angles C and D. That way, you'll have proven that all the sides have the same length and all the angles measure 90 degrees-- so the quadrilateral ABCD matches the definition of a square.
Hope this helped!
