Question

I am willing to work. I want to understand and contribute to the answer. Is triangle ABC with vertices A(-1, 4), B(3,1), and C(0,-3) a right triangle? Explain your answer using the slopes of the segments that form the sides of the triangle. I will include the visual document once the thread is open.

Answer 1

Did you find the slopes?

Answer 2

I'm currently working on that right now, we can go through each slope together, or you could wait a few minutes for me to finish up, sorry.

Answer 3

You have to find all the slopes of each segment. If two of those slopes are perpendicular to each other, then you know itʻs a right triangle

Answer 4

Perpendicular means it creates a 90 degree angle. You will know this if you find a slope that is an opposite reciprocal to another slope

Answer 5

For example

3 --> -1/3

2 ---> -1/2

Answer 6

Ok, I see what your saying. If you don't mind, could we possible first go over the finding the slope for each vertices together?

Answer 7

Do you know the formula that you would use?

Answer 8

There is a quick way of doing this, and then there is the harder way

Answer 9

I know in Algebra we use the m = Rise/Run, but I since this is Geometry wouldn't we be finding the medians of a triangle (vertices)?

Answer 10

For example, (xM, yM)...

Answer 11

The formula that you would use is

\[m = y _{2} - y _{1} \over x _{2} - x _{1}\]

Answer 12

You would just throw the vertices in, and find the slope of each length, one by one

Answer 13

That is the harder and longer way, yet it may be more clear to you.

Would you want me to show you the easier way?

Answer 14

The easier way is better for tests, and quickly moving through problems

Answer 15

I would like to observe the easier way, but could I also see the 'original' way as well later on?

Answer 16

Sure

Answer 17

Well, which way would you like to cover first? I would like to aid you in anyway possible.

Answer 18

Showing the easier way first

Answer 19

Ok so what you want to do is observe the triangle, and know what youʻre looking for.

Youʻre looking for a 90 degree angle to prove that this is a right triangle.

We know that it is a right triangle if itʻs the slopes of two segments, that are connected to the angle, are perpendicular to each other

Answer 20

For this, you can take advantage of the fact that this triangle is on a grid. You can count the units and determine the slopes easily without a formula.

Answer 21

So I counted the units, and this is what I got

Answer 22

Let me tell you how I got that

Answer 23

So counting the units isnʻt too hard, but, you can count 7 units easily, how do you know it is negative?

This is simple



Notice how that line looks like it is angling down, like itʻs falling. Whenever you see a line like this, you know it has a negative slope.

The slope of -7 is very steep, but notice how it follows that same motion, of going from the top left to the bottom right

Answer 24

Positive slopes, like 4/3, look like they are rising up. Btw, this is why itʻs called slope...like the slope of a mountain. They can be steep, or rise slowly.



Positive slope

Itʻs goes from the bottom left, to the top right

Rising up like a mountain

Positive slopes will look similar, but remember, like the -7 slopes, they can be steep like this



If ever unsure, use the slope formula, especially if your grade is counting on it

Answer 25

Any questions so far?

Answer 26

No, I believe I got. Just to make sure, this was the easy way right?

Answer 27

Yeah, you just have to make these observations, notice them, and itʻs just cake

Answer 28

Iʻm just explaining why these observations work, so you understand it. So you wonʻt make any mistakes

Answer 29

I've actually done it that way before, thank you for explaining it! Could we possible go over the formal way of doing it, if you have time?

Answer 30

Yeah, no problem

Answer 31

So the more formal way is using the slope formula

Remember it!

\[m = y _{2} - y _{1} \over x _{2} - x _{1} \]

Answer 32

The line doesnʻt go under the m, donʻt write it like that, idk why itʻs doing that



Like that