Geometric shapes are mathematical shapes that include squares and triangles
<h3>How to prove the shape is a square</h3>
The side lengths of a square are congruent, and the adjacent sides are perpendicular.
So, we start by calculating the side lengths using the following distance formula
![d = \sqrt{(x_1 -x_2)^2 + (y_1 -y_2)^2}](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7B%28x_1%20-x_2%29%5E2%20%2B%20%28y_1%20-y_2%29%5E2%7D)
Using the above formula, we have:
![AB = \sqrt{(3-2)^2 + (4+2)^2} = \sqrt{37](https://tex.z-dn.net/?f=AB%20%3D%20%5Csqrt%7B%283-2%29%5E2%20%2B%20%284%2B2%29%5E2%7D%20%3D%20%5Csqrt%7B37)
![BC = \sqrt{(2+4)^2 + (-2+1)^2} = \sqrt{37](https://tex.z-dn.net/?f=BC%20%3D%20%5Csqrt%7B%282%2B4%29%5E2%20%2B%20%28-2%2B1%29%5E2%7D%20%3D%20%5Csqrt%7B37)
![CD = \sqrt{(-4+3)^2 + (-1-5)^2} = \sqrt{37}](https://tex.z-dn.net/?f=CD%20%3D%20%5Csqrt%7B%28-4%2B3%29%5E2%20%2B%20%28-1-5%29%5E2%7D%20%3D%20%5Csqrt%7B37%7D)
![DA = \sqrt{(-3-3)^2 + (5-4)^2} = \sqrt{37}](https://tex.z-dn.net/?f=DA%20%3D%20%5Csqrt%7B%28-3-3%29%5E2%20%2B%20%285-4%29%5E2%7D%20%3D%20%5Csqrt%7B37%7D)
The above shows that the side lengths of the square are congruent.
Next, calculate the slope of the sides using:
![m = \frac{y_2 -y_1}{x_2 -x_1}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By_2%20-y_1%7D%7Bx_2%20-x_1%7D)
So, we have:
![m_{AB} = \frac{4 +2}{3 -2} = 6](https://tex.z-dn.net/?f=m_%7BAB%7D%20%3D%20%5Cfrac%7B4%20%2B2%7D%7B3%20-2%7D%20%3D%206)
![m_{BC} = \frac{-2 + 1}{2+4} = -\frac 16](https://tex.z-dn.net/?f=m_%7BBC%7D%20%3D%20%5Cfrac%7B-2%20%2B%201%7D%7B2%2B4%7D%20%3D%20-%5Cfrac%2016)
![m_{CD} = \frac{-1 -5}{-4+3} = 6](https://tex.z-dn.net/?f=m_%7BCD%7D%20%3D%20%5Cfrac%7B-1%20-5%7D%7B-4%2B3%7D%20%3D%206)
![m_{DA} = \frac{5 -4}{-3-3} = -\frac 16](https://tex.z-dn.net/?f=m_%7BDA%7D%20%3D%20%5Cfrac%7B5%20-4%7D%7B-3-3%7D%20%3D%20-%5Cfrac%2016)
Notice that the opposite slopes are congruent, and the adjacent slopes are opposite reciprocal.
The above highlight, and the equal side lengths show that the figure (1) is a square
<h3>How to prove the shape is a right isosceles triangle</h3>
The legs of a right isosceles triangle are congruent, and the legs are perpendicular.
So, we start by calculating the lengths of the legs using the following distance formula
![d = \sqrt{(x_1 -x_2)^2 + (y_1 -y_2)^2}](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7B%28x_1%20-x_2%29%5E2%20%2B%20%28y_1%20-y_2%29%5E2%7D)
Using the above formula, we have:
![XY = \sqrt{(5-4)^2 + (-1-4)^2} = \sqrt{26](https://tex.z-dn.net/?f=XY%20%3D%20%5Csqrt%7B%285-4%29%5E2%20%2B%20%28-1-4%29%5E2%7D%20%3D%20%5Csqrt%7B26)
![XZ = \sqrt{(5-0)^2 + (-1+2)^2} = \sqrt{26](https://tex.z-dn.net/?f=XZ%20%3D%20%5Csqrt%7B%285-0%29%5E2%20%2B%20%28-1%2B2%29%5E2%7D%20%3D%20%5Csqrt%7B26)
The above shows that the legs of the right isosceles triangle are congruent.
Next, calculate the slope of the legs using:
![m = \frac{y_2 -y_1}{x_2 -x_1}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By_2%20-y_1%7D%7Bx_2%20-x_1%7D)
So, we have:
![m_{XY} = \frac{5-4}{-1 -4} = -\frac{1}{5}](https://tex.z-dn.net/?f=m_%7BXY%7D%20%3D%20%5Cfrac%7B5-4%7D%7B-1%20-4%7D%20%3D%20-%5Cfrac%7B1%7D%7B5%7D)
![m_{XZ} = \frac{5-0}{-1+2} = 5](https://tex.z-dn.net/?f=m_%7BXZ%7D%20%3D%20%5Cfrac%7B5-0%7D%7B-1%2B2%7D%20%3D%205)
Notice that the slopes are opposite reciprocal.
The above highlight, and the equal legs show that the figure (2) is a right isosceles triangle
Read more about geometric shapes at:
brainly.com/question/14285697