Question

No sistema de coordenadas cartesianas a seguir está representado o triângulo ABC. Calcule a medida da altura relativa ao vértice C.

Answer 1

u.c => Unidade de comprimento

A(1,3)
B(7,1)

Encontrando a equação da reta AB

$m= \dfrac{(y_B-y_A}{(x_B-x_A)}\\\\\\m= \dfrac{1-3}{7-1}\\\\\\m= \dfrac{-2}{6}\\\\\\m= -\dfrac{1}{3}\\\\\\y-y_O=m\cdot(x-x_O)\\\\\\y-3= -\dfrac{1}{3}\cdot(x-1)\\\\\\y-3= -\dfrac{1}{3}x+\dfrac{1}{3}\\\\\\3y-9=-x+1\\\\\\x+3y-10=0$

Encontrando a distância da reta até o ponto C

$d= \dfrac{\bigg|ax_{O}+by_{O}+c\bigg|}{ \sqrt{a^2+b^2}}$

$d= \dfrac{\bigg|1\cdot(3)+(3)\cdot(5)-10\bigg|}{ \sqrt{(1)^2+(3)^2}}\\\\\\d=\dfrac{\bigg|3+15-10\bigg|}{ \sqrt{1+9}}\\\\\\d= \dfrac{8}{ \sqrt{10}}\cdot \dfrac{ \sqrt{10}}{ \sqrt{10}}\\\\\\d= \dfrac{8\cdot\sqrt{10} }{10}\\\\\\\boxed{d= \dfrac{4\cdot \sqrt{10}}{5}\ u.c}$