Answer:
a) Perimeter ![= {\bf 8} + \sqrt {\bf 113} + \sqrt {\bf 65}](https://tex.z-dn.net/?f=%3D%20%7B%5Cbf%208%7D%20%2B%20%5Csqrt%20%7B%5Cbf%20113%7D%20%2B%20%5Csqrt%20%7B%5Cbf%2065%7D%20)
b)Area ![={\bf 96}](https://tex.z-dn.net/?f=%3D%7B%5Cbf%2096%7D%20)
Step-by-step explanation:
Given ABC is a triangle with vertices at A(-2,-3), B(6,-3) and C(-1,5)
The vertices A(-2,-3), B(6,-3) and C(-1,5) are represented by
respectively
Now find the perimeter of the triangle ABC
The perimeter is found by first finding the three distances between the three vertices
given by
![d_{AB} = \sqrt {(x_A - x_B)^2 + (y_A - y_B)^2)}](https://tex.z-dn.net/?f=d_%7BAB%7D%20%3D%20%5Csqrt%20%7B%28x_A%20-%20x_B%29%5E2%20%2B%20%28y_A%20-%20y_B%29%5E2%29%7D)
![d_{BC} = \sqrt {(x_B - x_C)^2 + (y_B - y_C)^2)}](https://tex.z-dn.net/?f=d_%7BBC%7D%20%3D%20%5Csqrt%20%7B%28x_B%20-%20x_C%29%5E2%20%2B%20%28y_B%20-%20y_C%29%5E2%29%7D)
![d_{CA}= \sqrt {(x_C - x_A)^2 + (x_C - y_A)^2}](https://tex.z-dn.net/?f=d_%7BCA%7D%3D%20%5Csqrt%20%7B%28x_C%20-%20x_A%29%5E2%20%2B%20%28x_C%20-%20y_A%29%5E2%7D)
The perimeter is given by
Perimeter ![=d_{AB} + d_{BC} + d_{CA}](https://tex.z-dn.net/?f=%3Dd_%7BAB%7D%20%2B%20d_%7BBC%7D%20%2B%20d_%7BCA%7D)
now find ![d_{AB} = \sqrt {(x_A - x_B)^2 + (y_A - y_B)^2)}](https://tex.z-dn.net/?f=d_%7BAB%7D%20%3D%20%5Csqrt%20%7B%28x_A%20-%20x_B%29%5E2%20%2B%20%28y_A%20-%20y_B%29%5E2%29%7D)
![d_{AB}= \sqrt {(-2 - 6)^2 + (-3+3 )^2)}](https://tex.z-dn.net/?f=d_%7BAB%7D%3D%20%5Csqrt%20%7B%28-2%20-%206%29%5E2%20%2B%20%28-3%2B3%20%29%5E2%29%7D)
![d_{AB} = \sqrt {(-8)^2 + (0)^2)}](https://tex.z-dn.net/?f=d_%7BAB%7D%20%3D%20%5Csqrt%20%7B%28-8%29%5E2%20%2B%20%280%29%5E2%29%7D)
![d_{AB} = \sqrt {8^2}](https://tex.z-dn.net/?f=d_%7BAB%7D%20%3D%20%5Csqrt%20%7B8%5E2%7D)
![d_{AB}= \sqrt {64}](https://tex.z-dn.net/?f=d_%7BAB%7D%3D%20%5Csqrt%20%7B64%7D)
![d_{AB} = 8](https://tex.z-dn.net/?f=d_%7BAB%7D%20%3D%208)
Similarly we find ![d_{BC} = \sqrt {(x_B - x_C)^2 + (y_B - y_C)^2)}](https://tex.z-dn.net/?f=d_%7BBC%7D%20%3D%20%5Csqrt%20%7B%28x_B%20-%20x_C%29%5E2%20%2B%20%28y_B%20-%20y_C%29%5E2%29%7D)
![d_{BC}= \sqrt {(6+1)^2 + (-3-5)^2)}](https://tex.z-dn.net/?f=d_%7BBC%7D%3D%20%5Csqrt%20%7B%286%2B1%29%5E2%20%2B%20%28-3-5%29%5E2%29%7D)
![d_{BC} = \sqrt {(7)^2 + (-8)^2)}](https://tex.z-dn.net/?f=d_%7BBC%7D%20%3D%20%5Csqrt%20%7B%287%29%5E2%20%2B%20%28-8%29%5E2%29%7D)
![d_{BC}= \sqrt {49 + 64}](https://tex.z-dn.net/?f=d_%7BBC%7D%3D%20%5Csqrt%20%7B49%20%2B%2064%7D)
![d_{BC} = \sqrt {113}](https://tex.z-dn.net/?f=d_%7BBC%7D%20%3D%20%5Csqrt%20%7B113%7D)
find ![d_{CA} = \sqrt {(x_C - x_A)^2 + (x_C - y_A)^2}](https://tex.z-dn.net/?f=d_%7BCA%7D%20%3D%20%5Csqrt%20%7B%28x_C%20-%20x_A%29%5E2%20%2B%20%28x_C%20-%20y_A%29%5E2%7D)
![d_{CA} = \sqrt {(-1 +2)^2 + (5+3)^2}](https://tex.z-dn.net/?f=d_%7BCA%7D%20%3D%20%5Csqrt%20%7B%28-1%20%2B2%29%5E2%20%2B%20%285%2B3%29%5E2%7D)
![d_{CA} = \sqrt {(1)^2 + (8)^2}](https://tex.z-dn.net/?f=d_%7BCA%7D%20%3D%20%5Csqrt%20%7B%281%29%5E2%20%2B%20%288%29%5E2%7D)
![d_{CA} = \sqrt {1 + 64}](https://tex.z-dn.net/?f=d_%7BCA%7D%20%3D%20%5Csqrt%20%7B1%20%2B%2064%7D)
![d_{CA} = \sqrt {65}](https://tex.z-dn.net/?f=d_%7BCA%7D%20%3D%20%5Csqrt%20%7B65%7D)
Now adding the distances we get
Perimeter ![=d_{AB}+ d_{BC} + d_{CA}](https://tex.z-dn.net/?f=%3Dd_%7BAB%7D%2B%20d_%7BBC%7D%20%2B%20d_%7BCA%7D)
Perimeter ![= 8+ \sqrt {113} + \sqrt {65}](https://tex.z-dn.net/?f=%3D%208%2B%20%5Csqrt%20%7B113%7D%20%2B%20%5Csqrt%20%7B65%7D)
b) Area of the given triangle ABC
The formula for the area of the triangle defined by the three vertices A, B and C is given by:
![Area= \frac{1}{2} {\det {\left[\begin{array}{ccc}x_A&x_B&x_C\\y_A&y_B&y_C\\1&1&1\end{array}\right]}}](https://tex.z-dn.net/?f=Area%3D%20%5Cfrac%7B1%7D%7B2%7D%20%7B%5Cdet%20%7B%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx_A%26x_B%26x_C%5C%5Cy_A%26y_B%26y_C%5C%5C1%261%261%5Cend%7Barray%7D%5Cright%5D%7D%7D)
where det is the determinant of the three by three matrix.
![Area=\frac{1}{2}{{\det \left[\begin{array}{ccc}-2&6&-1\\ -3& -3&5\\ 1 & 1 & 1\end{array}\right]}}](https://tex.z-dn.net/?f=Area%3D%5Cfrac%7B1%7D%7B2%7D%7B%7B%5Cdet%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-2%266%26-1%5C%5C%20-3%26%20-3%265%5C%5C%201%20%26%201%20%26%201%5Cend%7Barray%7D%5Cright%5D%7D%7D)
![Area=\frac{1}{2}[-2(-3-5)-6(-3-5)-1(-3+3)+3(6+1)-3(-2+1)-5(-2-6)+1(30-3)-1(-10-3)+1(6+18)]](https://tex.z-dn.net/?f=Area%3D%5Cfrac%7B1%7D%7B2%7D%5B-2%28-3-5%29-6%28-3-5%29-1%28-3%2B3%29%2B3%286%2B1%29-3%28-2%2B1%29-5%28-2-6%29%2B1%2830-3%29-1%28-10-3%29%2B1%286%2B18%29%5D%20)
![Area=\frac{1}{2}[-2(-8)-6(-8)-1(0)+3(7)-3(-1)-5(-8)+1(27)-1(-13)+1(24)]](https://tex.z-dn.net/?f=Area%3D%5Cfrac%7B1%7D%7B2%7D%5B-2%28-8%29-6%28-8%29-1%280%29%2B3%287%29-3%28-1%29-5%28-8%29%2B1%2827%29-1%28-13%29%2B1%2824%29%5D%20)
![Area=\frac{1}{2}[16+48+0+21+3+40+27+13+24]](https://tex.z-dn.net/?f=Area%3D%5Cfrac%7B1%7D%7B2%7D%5B16%2B48%2B0%2B21%2B3%2B40%2B27%2B13%2B24%5D)
![Area=\frac{1}{2} (192)](https://tex.z-dn.net/?f=Area%3D%5Cfrac%7B1%7D%7B2%7D%20%28192%29)
![Area=96](https://tex.z-dn.net/?f=Area%3D96)