Calcular el área y perimetro de poligonos en un plano cartesianoA( - 6, - 7), B ( - 5, 2) , C(6, 2) Y D (8, - 7)?

Primero ubicamos los puntos en el plano cartesiano y unimos los puntos para ver que figura forma (en la imagen te lo dejo)

Perímetro

Para hallar el perímetro debemos saber la distancia entre los puntos, para ello utilizamos la sgte formula :

<img src="https://tex.z-dn.net/?f=d%3D%20%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D%20%0A" />

Ahora hallemos las medidas de los lados

<img src="https://tex.z-dn.net/?f=%5Coverline%7BAB%7D%0A%5C%5CA%28-6%2C-7%29%20%5C%20y%20%5C%20B%20%28-5%2C2%29%0A%5C%5C%5C%5Cd%3D%20%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%28%28-5%29-%28-6%29%29%5E2%2B%28%282%29-%28-7%29%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B-5%2B6%29%5E2%2B%282%2B7%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%281%29%5E2%2B%289%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B1%2B81%7D%0A%5C%5Cd%3D%20%5Csqrt%7B82%7D%20%0A%5C%5Cd%3D9.05" />

<img src="https://tex.z-dn.net/?f=%5Coverline%7BBC%7D%0A%5C%5C%20%20B%20%28-5%2C2%29%20%5C%20y%20%5C%20C%286%2C%202%29%0A%5C%5C%5C%5Cd%3D%20%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%28%286%29-%28-5%29%29%5E2%2B%28%282%29-%282%29%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%286%2B5%29%5E2%2B%282-2%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%2811%29%5E2%2B%280%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%2811%29%5E2%7D%0A%5C%5Cd%3D%2011" />

<img src="https://tex.z-dn.net/?f=%5Coverline%7BCD%7D%0A%5C%5CC%286%2C%202%29%20%5C%20y%20%5C%20D%20%288%2C-7%29%0A%5C%5C%5C%5Cd%3D%20%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D%0A%5C%5C%20d%3D%20%5Csqrt%7B%28%288%29-%286%29%29%5E2%2B%28%28-7%29-%282%29%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%288-6%29%5E2%2B%28-7-2%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%282%29%5E2%2B%28-9%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B4%2B81%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%2885%7D%0A%5C%5Cd%20%3D%209.22" />

<img src="https://tex.z-dn.net/?f=%5Coverline%7BDA%7D%0A%5C%5C%20D%20%288%2C-7%29%20%5C%20y%20%5C%20A%28-6%2C-7%29%0A%5C%5C%5C%5Cd%3D%20%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%28%28-6%29-%288%29%29%5E2%2B%28%28-7%29-%28-7%29%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%28-6-8%29%5E2%2B%28-7%2B7%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B%28-14%29%5E2%2B%280%29%5E2%7D%0A%5C%5Cd%3D%20%5Csqrt%7B196%7D%0A%5C%5Cd%3D14%0A" />

P = 14 + 9.

22 + 11 + 9.

05

P = 43, 27

Área

La figura es un trapecio, el area del trapecio es igual a :

<img src="https://tex.z-dn.net/?f=A%3D%20%5Cfrac%7BBase%5C%20mayor%2BBase%20%5C%20menor%7D%7B2%7D%2Ah%0A%20" />

<img src="https://tex.z-dn.net/?f=Base%20%5C%20mayor%20%3D%20%5Coverline%7BAD%7D%0A%5C%5CBase%20%5C%20menor%20%3D%20%5Coverline%7BBC%7D" />

La altura podemos encontrarla de varias maneras en este caso solo contare los puntos que separan a la figura (en la imagen te lo muestro, es la linea de negro)

h = 9

Entonces :

<img src="https://tex.z-dn.net/?f=A%3D%20%5Cfrac%7BB%2Bb%7D%7B2%7D%2Ah%0A%5C%5C%5C%5CA%3D%20%5Cfrac%7B14%2B11%7D%7B2%7D%2A9%0A%5C%5C%5C%5CA%3D%20%5Cfrac%7B25%2A9%7D%7B2%7D%20%0A%5C%5C%5C%5CA%3D%20%5Cfrac%7B225%7D%7B2%7D%20%0A%5C%5C%5C%5CA%3D112%2C5" />.