Los vertices de un triangulo rectangulo son los puntos A (1, 2), B (4, - 2) y C (4, 2)?

Solución

Se tienen los siguientes datos :

<img src="https://tex.z-dn.net/?f=A%3D%281%2C2%29%5C%5CB%3D%284%2C-2%29%5C%5CC%3D%284%2C2%29" />

Luego aplicando la fórmula de la distancia para encontrar las longitudes de.

<img src="https://tex.z-dn.net/?f=a" /> , <img src="https://tex.z-dn.net/?f=b" /> y <img src="https://tex.z-dn.net/?f=c" /> donde : <img src="https://tex.z-dn.net/?f=d_%7Bc%7D%3Dd_%7BAB%7D%5C%5Cd_%7Ba%7D%3Dd_%7BAC%7D%5C%5Cd_%7Bb%7D%3Dd_%7BCB%7D" /> se obtiene :

<img src="https://tex.z-dn.net/?f=d_%7Bc%7D%3D%5Csqrt%7B%284-1%29%5E%7B2%7D%2B%28-2-2%29%5E%7B2%7D%7D%5C%5Cd_%7Bc%7D%3D%5Csqrt%7B%283%29%5E%7B2%7D%2B%28-4%29%5E%7B2%7D%7D%5C%5Cd_%7Bc%7D%3D%5Csqrt%7B9%2B16%7D%5C%5Cd_%7Bc%7D%3D%5Csqrt%7B25%7D%5C%5Cd_%7Bc%7D%3D5%5C%5Cd_%7Ba%7D%3D%5Csqrt%7B%284-1%29%5E%7B2%7D%2B%282-2%29%5E%7B2%7D%7D%5C%5Cd_%7Ba%7D%3D%5Csqrt%7B%283%29%5E%7B2%7D%2B0%7D%5C%5Cd_%7Ba%7D%3D%5Csqrt%7B9%7D%5C%5Cd_%7Ba%7D%3D3%5C%5Cd_%7Bb%7D%3D%5Csqrt%7B%284-4%29%5E%7B2%7D%2B%282-%28-2%29%29%5E%7B2%7D%7D%5C%5Cd_%7Bb%7D%3D%5Csqrt%7B0%2B%284%29%5E%7B2%7D%7D%5C%5Cd_%7Bb%7D%3D%5Csqrt%7B16%7D%5C%5Cd_%7Bb%7D%3D4" />

Finalmente busquemos el área y nos queda :

<img src="https://tex.z-dn.net/?f=A%3D%5Cfrac%7Bbh%7D%7B2%7D%5C%5CA%3D%5Cfrac%7B%28d_%7BAC%7D%29%28d_%7BCB%7D%29%7D%7B2%7D%3D%5Cfrac%7B%28d_%7Ba%7D%29%28d_%7Bb%7D%29%7D%7B2%7D%5C%5CA%3D%5Cfrac%7B%283%29%284%29%7D%7B2%7D%5C%5CA%3D6%5C%2Cu%5E%7B2%7D" />

Saludos.