How would you show that a triangle with vertices (13,-2), (9,-8), (5,-2) is isosceles?

Question

Sadie Adams · Accepted Answer

#color(crimson)("The distances AB and BC are " sqrt52 " and therefore isosceles."#

If we label them A(13,-2) B(9-8) and C(5-2)

The distance between A and B: #13=>9=4# #-2=>-8=6# #4^2+6^2=52# so the distance is #sqrt52#

The distance between B and C: #9=>5=4# #-8=>-2=6# #4^2+6^2=52# the distance is #sqrt52#

The distance between A and C is 8 as they are both on -2 for #y#

#color(crimson)("The distances AB and BC are " sqrt52 " and therefore isosceles."#

Theodore Abad · Answer

see below

An isosceles triangle is one with two equal lengths. To find the lengths of the sides with coordinates #(x_1,y_1),(x_2,y_2)# we use Pythagoras' theorem #d=sqrt((x_2-x_2)^2+(y_2-y_1)^2)# from the information given #d_1= sqrt((13-9)^2 +(-2- -8)^2)# #color(red)(d_1=sqrt(4^2+6^2)=sqrt(52)--(1))# #d_2=sqrt((13-5)^2+(-2 --2)^2)# #color(red)(d_2=sqrt(8^2+0^2)=sqrt64=8--(2)# #d_3=sqrt((9-5)^2+(-8 --2 )^2# #color(red)(d_3=sqrt(4^2+6^2)=sqrt52---(3)# we see #d_1=d_3# since we have two equal sides the triangle with the given vertices is isosceles

Wyatt Anderson · Answer

Below

One way of proving that it is an isosceles triangle is by calculating the length of each side since two sides of equal lengths means that it is an isosceles triangle. Length of #(13,-2) & (9,-8)# =#sqrt((13-9)^2+(-2+8)^2)# =#sqrt(16+36)# =#sqrt52# =#2sqrt13# units Length of #(9,-8) & (5,-2)# =#sqrt((9-5)^2+(-8+2)^2)# =#sqrt(16+36)# =#sqrt52# =#2sqrt13# units Length of #(13,-2) & (5,-2)# =#sqrt((13-5)^2+(-2+2)^2)# =#sqrt64# =#8# units From the above calculations, you'll notice that length of #(13,-2) & (9,-8)# and length of #(9,-8) & (5,-2)# are the same. therefore, you can prove that the triangle is isosceles

Parker Allen · Answer

undefined To show that the triangle with vertices (13,-2), (9,-8), (5,-2) is isosceles, we can calculate the distances between the vertices using the distance formula. If two of the distances are equal, then the triangle is isosceles.

Let's denote the vertices as follows: A(13,-2), B(9,-8), and C(5,-2).

Then, we calculate the lengths of the sides:

AB = √[(13 - 9)² + (-2 - (-8))²] = √[16 + 36] = √52
BC = √[(9 - 5)² + (-8 - (-2))²] = √[16 + 36] = √52
AC = √[(13 - 5)² + (-2 - (-2))²] = √[64] = 8

Since AB = BC, the triangle is isosceles.