Question

Determine if triangle RST with coordinates R (3, 4), S (5, 5), and T (6, 1) is a right triangle. Use evidence to support your claim. If it is not a right triangle, what changes can be made to make it a right triangle? Be specific.

Answer 1

@Hero @nincompoop @robtobey @Compassionate

Answer 2

Hint:
https://www.desmos.com/calculator/fq8bdv3qo6

Answer 3

No, it's not a right triangle, it's an Isosceles. @Hero

Answer 4

Isosceles? Who said it was isosceles?
https://www.desmos.com/calculator/kp5ypkg6y6

Answer 5

It looks like an isosceles triangle. Instead of a right triangle.

Answer 6

In mathematics, you verify things using mathematical law and reasoning, because looks can be deceiving.

Answer 7

I'm still not sure how to solve the problem, though..@Hero

Answer 8

@phi

Answer 9

there a two ways to prove you have (or do not have) a right triangle:
1) show it has a right angle, or
2) show pythagoras works

Answer 10

I'm still not sure. @phi

Answer 11

It doesn't have a right angle, from what I can see.

Answer 12

to show it has a right angle, we use the idea that perpendicular lines have slopes that are *negative reciprocals*

for example, 2 and -½ , or ¾ and -4/3
(flip and then multiply by -1 to go from one slope to the other)

that means we need to find the slopes between points
R and S
S and T
T and R
and see if any pair is the negative reciprocal of the other

Answer 13

can you find the slope between R and S
R(3, 4), S (5, 5)
?

Answer 14

\[\frac{ 3-5 }{ 4-5 } = \frac{ -2 }{ -1}\]

Answer 15

close. but it is change in y divided by change in x
you did change in x divided by change in y

Answer 16

So, it'd be \[\frac{ -1 }{-2 }\]

Answer 17

yes. and minus divided by minus is plus

so just 1/2

Answer 18

okay. Then what?

Answer 19

now let's find the slope between S and T

Answer 20

S = (5, 5)
T = (6, 1)

\[\frac{ 5 - 1 }{ 5 - 6 } = \frac{ 4}{ -1}\]

Answer 21

yes. can you simplify that ?

Answer 22

I don't know how..

Answer 23

it has only 1 minus sign so the answer is negative
in other words
\[ \frac{4}{-1}= - \frac{4}{1} \]
now what is 4 divided by 1 ?

Answer 24

4.

Answer 25

so the answer is -4

so far we have
slope R to S = ½
slope S to T = -4
those are not negative reciprocals so the angle formed by RS and ST is not a right angle.
one more side to try
find the slope between R and T

Answer 26

R = (3, 4)
T = (6, 1)

\[\frac{ 4-1 }{ 3-6 } = \frac{ 3 }{ -3}\]

Answer 27

yes. now simplify it

Answer 28

Is it 1 ?

Answer 29

it has a minus sign (in the bottom) so the answer will be negative.
you get -1

Answer 30

here is what you know
negative reciprocal
slope R to S = ½ -2
slope S to T = -4 1/4
slope R to T = -1 1

there is no pair that form negative reciprocals, so you know none of the lines are perpendicular to each other. In other words, none of the lines cross at 90º
so this triangle does not have a 90 degree angle, so it is *not* a right triangle

Answer 31

Now for the hard part.
If it is not a right triangle, what changes can be made to make it a right triangle?

Answer 32

Our best bet is notice that
slope R to T = -1
the negative reciprocal is +1

so I would move point S
so that the slope from R to S is +1
that will make RS and RT perpendicular

Answer 33

so far we have
R(3, 4), S (5, 5)
the slope is
\[ \frac{5-4}{5-3} \]
any idea how to change S so that we get +1 ?

Answer 34

to get a 1, you want the top number and the bottom number to be the same, because anything divided by itself is 1

if we leave the top number 5-4=1
then you want the bottom number to be 1
we want to change 5-3 to n-3 where n is a new number, so that we get 1
any idea what n should be ?

Answer 35

@phi i need your help explaining the rest of this i got everything else so far!:)

Answer 36

@iambatman could you please help me on this?

Answer 37

@e.mccormick could you please help me on this?