Answer:
Part 2) Triangle ABC is a right triangle (see the explanation)
Part 3) Quadrilateral QRST is not a parallelogram (see the explanation)
Step-by-step explanation:
Part 2) we have
A (5, 2), B (2, 4), and C (7, 5)
Plot the figure to better understand the problem
see the attached figure
we know that
the formula to calculate the distance between two points is equal to
![d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28y2-y1%29%5E%7B2%7D%2B%28x2-x1%29%5E%7B2%7D%7D)
step 1
Find the length side AB
A (5, 2), B (2, 4)
substitute the values in the formula
![d=\sqrt{(4-2)^{2}+(2-5)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%284-2%29%5E%7B2%7D%2B%282-5%29%5E%7B2%7D%7D)
![d=\sqrt{(2)^{2}+(-3)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%282%29%5E%7B2%7D%2B%28-3%29%5E%7B2%7D%7D)
![d_A_B=\sqrt{13}\ units](https://tex.z-dn.net/?f=d_A_B%3D%5Csqrt%7B13%7D%5C%20units)
step 2
Find the length side BC
B (2, 4), C (7, 5)
substitute the values in the formula
![d=\sqrt{(5-4)^{2}+(7-2)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%285-4%29%5E%7B2%7D%2B%287-2%29%5E%7B2%7D%7D)
![d=\sqrt{(1)^{2}+(5)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%281%29%5E%7B2%7D%2B%285%29%5E%7B2%7D%7D)
![d_B_C=\sqrt{26}\ units](https://tex.z-dn.net/?f=d_B_C%3D%5Csqrt%7B26%7D%5C%20units)
step 3
Find the length side AC
A (5, 2), C (7, 5)
substitute the values in the formula
![d=\sqrt{(5-2)^{2}+(7-5)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%285-2%29%5E%7B2%7D%2B%287-5%29%5E%7B2%7D%7D)
![d=\sqrt{(3)^{2}+(2)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%283%29%5E%7B2%7D%2B%282%29%5E%7B2%7D%7D)
![d_A_C=\sqrt{13}\ units](https://tex.z-dn.net/?f=d_A_C%3D%5Csqrt%7B13%7D%5C%20units)
step 4
Verify if the triangle ABC is a right triangle
we know that a right triangle must satisfy the Pythagorean Theorem
so
![BC^2=AB^2+AC^2](https://tex.z-dn.net/?f=BC%5E2%3DAB%5E2%2BAC%5E2)
Remember that the hypotenuse is the greater side
substitute the values
![(\sqrt{26})^2=(\sqrt{13})^2+(\sqrt{13})^2](https://tex.z-dn.net/?f=%28%5Csqrt%7B26%7D%29%5E2%3D%28%5Csqrt%7B13%7D%29%5E2%2B%28%5Csqrt%7B13%7D%29%5E2)
![26=13+13](https://tex.z-dn.net/?f=26%3D13%2B13)
----> is true
therefore
Triangle ABC is a right triangle
Part 3) we have
Q (5, 1), R (8, 7), S (14, 10) and T (10, 2)
we know that
The opposite sides of a parallelogram are parallel and congruent
the formula to calculate the distance between two points is equal to
![d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28y2-y1%29%5E%7B2%7D%2B%28x2-x1%29%5E%7B2%7D%7D)
Step 1
Find the length side QR
Q (5, 1), R (8, 7)
substitute in the formula
![d=\sqrt{(7-1)^{2}+(8-5)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%287-1%29%5E%7B2%7D%2B%288-5%29%5E%7B2%7D%7D)
![d=\sqrt{(6)^{2}+(3)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%286%29%5E%7B2%7D%2B%283%29%5E%7B2%7D%7D)
![d_Q_R=\sqrt{45}\ units](https://tex.z-dn.net/?f=d_Q_R%3D%5Csqrt%7B45%7D%5C%20units)
Step 2
Find the length side RS
R (8, 7), S (14, 10)
substitute in the formula
![d=\sqrt{(10-7)^{2}+(14-8)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%2810-7%29%5E%7B2%7D%2B%2814-8%29%5E%7B2%7D%7D)
![d=\sqrt{(3)^{2}+(6)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%283%29%5E%7B2%7D%2B%286%29%5E%7B2%7D%7D)
![d_R_S=\sqrt{45}\ units](https://tex.z-dn.net/?f=d_R_S%3D%5Csqrt%7B45%7D%5C%20units)
Step 3
Find the length side ST
S (14, 10), T (10, 2)
substitute in the formula
![d=\sqrt{(2-10)^{2}+(10-14)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%282-10%29%5E%7B2%7D%2B%2810-14%29%5E%7B2%7D%7D)
![d=\sqrt{(-8)^{2}+(-4)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28-8%29%5E%7B2%7D%2B%28-4%29%5E%7B2%7D%7D)
![d_S_T=\sqrt{80}\ units](https://tex.z-dn.net/?f=d_S_T%3D%5Csqrt%7B80%7D%5C%20units)
Step 4
Find the length side QT
Q (5, 1), T (10, 2)
substitute in the formula
![d=\sqrt{(2-1)^{2}+(10-5)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%282-1%29%5E%7B2%7D%2B%2810-5%29%5E%7B2%7D%7D)
![d=\sqrt{(1)^{2}+(5)^{2}}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%281%29%5E%7B2%7D%2B%285%29%5E%7B2%7D%7D)
![d_Q_T=\sqrt{26}\ units](https://tex.z-dn.net/?f=d_Q_T%3D%5Csqrt%7B26%7D%5C%20units)
Step 5
Compare the length of the opposite sides
QR and ST
![\sqrt{45}\ units \neq \sqrt{80}\ units](https://tex.z-dn.net/?f=%5Csqrt%7B45%7D%5C%20units%20%5Cneq%20%5Csqrt%7B80%7D%5C%20units)
![d_Q_R \neq d_S_T](https://tex.z-dn.net/?f=d_Q_R%20%5Cneq%20d_S_T)
RS and QT
![\sqrt{45}\ units \neq \sqrt{26}\ units](https://tex.z-dn.net/?f=%5Csqrt%7B45%7D%5C%20units%20%5Cneq%20%5Csqrt%7B26%7D%5C%20units)
![d_R_S \neq d_Q_T](https://tex.z-dn.net/?f=d_R_S%20%5Cneq%20d_Q_T)
Opposite sides are not congruent
therefore
Quadrilateral QRST is not a parallelogram