Question 1203124

A triangle is formed by the vertices A({{{1}}}, {{{2}}}), B({{{-5}}}, {{{-3}}}), and C({{{1}}},{{{-4}}})


a.) Classify this triangle


check the distance between {{{AB}}}, {{{AC}}}, and {{{BC}}}

{{{AB}}}

*[invoke formula_distance 1, 2, -5, -3]

{{{AB=7.8}}}



{{{AC}}}

 *[invoke formula_distance 1, 2, 1, -4]

{{{AC=6}}}



and {{{BC}}}

*[invoke formula_distance -5, -3, 1, -4]

{{{BC=6.08}}}



so, the lengths of the sides are different and we have a {{{scalene}}} triangle 


b.) Identify the centroid using the intersection of medians


find  midpoints:

A({{{1}}}, {{{2}}}), B({{{-5}}}, {{{-3}}}) => ({{{x1}}},{{{y1}}})=({{{(1-5)/2}}}, {{{(2-3)/2)=({{{-2, {{{-1/2)


A({{{1}}},{{{ 2}}}),  C({{{1}}},{{{-4}}})=> ({{{x2}}},{{{y2}}})= ({{{(1+1)/2}}}, {{{(2-4)/2}}})=({{{1}}}, {{{-1}}})


 B({{{-5}}}, {{{-3}}}), C({{{1}}},{{{-4}}})=> ({{{x3}}},{{{y3}}})= ({{{(-5+1)/2}}}, {{{(-3-4)/2}}})=({{{-2}}}, {{{-7/2}}})


find equations of the lines passing through the points ({{{-2}}}, {{{-1/2}}}) and ({{{1}}}, {{{-4}}}), as through the points ({{{-5}}}, {{{-3}}}) and ({{{1}}},{{{ -1}}})


*[invoke calculating_slope -2, "-1/2", 1, -4]

*[invoke calculating_slope -5, -3, 1, -1] 



{{{ y=(-7/6)x-17/6}}}
{{{y=(1/3)x-4/3}}}


find intersection:

*[invoke solving_linear_system_by_elimination "7/6", 1, "-17/6", "-1/3", 1, "-4/3"



intersection is at: ({{{-1}}},{{{-5/3}}}) which is centroid



c.) There is a formula for centroid, which is (x1+x2+x3 ÷ 3 , y1+y2+y3 ÷ 3), calculate the 

centroid using this formula. Do your answers match?

({{{(x1+x2+x3)/ 3}}} , {{{(y1+y2+y3)/ 3}}})=({{{(-2+1+-2)/ 3 }}},{{{ (-1/2-1-7/2)/ 3}}})= ({{{-1}}},{{{-5/3}}}) =>answers match