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geometry: please help w/ how to calculate orthocenters, circumcenters, and centroids

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I was helping my ds with math last night, but we went to bed tired and frustrated with these sorts of problems.  I think we might have been okay with circumcenter, but not the other two.  The book was not helpful on this.  We plan to try again when fresher on Saturday morning.

 

 

1.  Given triangle ABC with coordinates: A (-2,1) B (0, -2) C (5, -2)  find the coordinates for the centroid, orthocenter, and circumcenter.

 

2.   Given triangle ABC with coordinates: A (2, -3) B (3, 3)  C (-1, -2)     find same as in #1.

 

 

 

 

 

It seems like we need to be able to use:  Point slope formula, (y-y1)=m(x-x1) 

 

and then set two equations equal to each other to get the intersection.

 

E.g.,  by solving for two altitude equations, get the orthocenter ; or for  two perpendicular bisector equations, get the circumcenter;   or for two lines passing through 2 medians and 2 vertices get the centroid.   But we can't seem to do it.  We tried and came up with answers that clearly are wrong based on an approximation from drawing it on graph paper.

 

Are we on the right track as to what we think we need to do, and maybe were just too tired to do it?  Or are we off track in what we are trying to do?

 

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I was going to try to show, if I can do it in type, what we actually did, so that maybe someone would find the errors, but it went to school with ds in his notebook.

 

Anyway, we had started with finding all the side slopes, I think correctly, and then slopes perpendicular to those, also I think correctly.

 

We also identified midpoints of the sides, I think correctly.

 

So I am guessing that either we are off on thinking we need to use Point slope formula, (y-y1)=m(x-x1), or that we entered info wrong into the formula, or made errors when we tried to solve with the formulas for two intersecting lines set equal to each other.

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Circumcenter example is on page 4 of 10 http://lph.lpisd.org/ourpages/auto/2017/1/3/43202230/Lesson%206_2.pdf

Centroid example is on page 3 of 8 and Orthocenter example is on 4 of 8 (the next page) http://lph.lpisd.org/ourpages/auto/2017/1/3/51067874/Lesson%206_3.pdf

 

I’m half asleep at the library waiting for my kids’ chinese tutor. I’ll try working out your problem later.

 

ETA:

Question 1, circumcenter is (5/2, 11/6)

Slope of BC is 0

Slope of AB is -3/2

Midpoint of BC is (5/2, -2)

Midpoint of AB is (-1, -1/2)

 

Sum the x co-ordinates and divide by 3, sum the y co-ordinates and divide by 3

Centroid is (1,-1)

 

y = -2 is perpendicular drop from A to BC

y = x - 16/3 is perpendicular drop from C to AB

Solve y = -2 and y = x - 16/3

Orthocenter is (-2, -20/3)

 

Problem 2

Centroid is (4/3, -2/3)

 

Midpoint of AB is (5/2, 0)

Midpoint of BC is (1, 1/2)

Midpoint of AC is (1/2, -5/2)

 

Slope of AB is 6

Slope of BC is 5/4

Slope of AC is -1/3

Edited by Arcadia

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How did you find those pdfs?

Google, Bing, DuckDuckGo :lol: my younger boy needs lots of supplements for all subjects so I have lots of practice web surfing

 

Problem 2’s orthocenter is more tedious to compute since all of the triangle’s sides are slanted (not horizontal nor vertical) unlike Problem 1 which has a horizontal side.

 

I did use GeoGebra online (there is a GeoGebra app as well) to double check in case of computation errors. I don’t have graph paper on hand to double check by drawing.

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Google, Bing, DuckDuckGo :lol: my younger boy needs lots of supplements for all subjects so I have lots of practice web surfing

 

Problem 2’s orthocenter is more tedious to compute since all of the triangle’s sides are slanted (not horizontal nor vertical) unlike Problem 1 which has a horizontal side.

 

I did use GeoGebra online (there is a GeoGebra app as well) to double check in case of computation errors. I don’t have graph paper on hand to double check by drawing.

 

 

I'd tried a lot of Googling, but got nothing close to that good.  My terms were things like "how to calculate orthocenter of triangle"  or formula centroid triangle.  Ds had also tried last night.  

 

First I've heard of Geogebra.  Off to take a look!

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This morning I got one (orthocenter for problem 1) to come out correctly and am realizing that for both ds and myself going over prior attempts, even though we are using graph paper to try to keep things clear and well ordered, we made mistakes in putting in a wrong co-ordinate point at some stage, or using a wrong slope figure, or forgetting to distribute to both the x and x1 , or losing a negative sign...

 

Ds has some dyslexia which usually has not affected math much as long as he used graph paper, but maybe is now.  I have had a traumatic brain injury and this many steps and things to keep in order and clear seems to be beyond what I can currently do readily.

 

I'm thinking maybe for both me and ds when we try again tomorrow, to try colored pencils for parentheses, negatives, and so on.   Any ideas?

 

Does it tend to be clearer for people to work out all the side slopes first and then the perpendicular slopes etc., or to do each problem separately?  I had us try all sides, calculate all midpoints, etc.,  first, thinking it would save steps, and thinking that is how I would have done when in geometry myself years ago.

 

But we had errors like putting in the slope for a perpendicular rather than the slope for the vertex to median needed to find the centroid as one error made at one point. So maybe we need to do each problem on its own step-by-step?

 

This morning I named the midpoints Q, R, and S, which seemed to help as compared to previously calling them MAB; MBC; MAC,   which along with mAB, mBC, ...  seems to have been part of the confusion and errors problem.  I also named the orthocenter O.  I think maybe calling the Centroid D and the Circumcenter E could help.  Though I can already feel my thinking starting to feel like that is too many letters to keep straight.

 

Maybe this is a Learning Challenges issue, not a High School Board issue.

 

 

?????

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I drew a labeled diagram for each part of the problem so I had three diagrams for problem 1. After each diagram, I wrote down the values I need for example the midpoints and the slopes. Then I worked it out.

 

Problem 1’s circumcenter took me about 30mins working out and double checking in my caffeine deficit mode. My night owl also makes mistakes when sleepy, he just woke up.

 

I like my geometry diagrams in colors as I prefer visuals when doing math or science. My kids have a BIC four-color ballpoint pen each as it helps them when they need color.

 

ETA:

After drawing each diagram, I write down what I am given (coordinates) and what I need to compute (midpoints, slopes, etc). Then I just concentrate on solving that part of the problem. That’s so that I don’t have too much info in my temporary storage/working memory.

Edited by Arcadia

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Off to the side, or for a nasty problem, on another sheet of paper, you should label every single variable with what it signifies, in words. Then, when you find out the numerical value, put it by the variable. e.g. Q = midpoint of AB. Then when you get tangled up with letters, you can look back to your reference sheet, and you don't have to scan through your paper looking for the numbers. 

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Sum the x co-ordinates and divide by 3, sum the y co-ordinates and divide by 3

Centroid is (1,-1)

 

 

 

 Whoa!  I woke in wee hours realizing what you were saying!  For Centroid:

 

You're summing all 3 of the x values and dividing by 3, and same for y values?

 

I see that works for these!  Does that always work?  

 

We started by using the equations for 2 lines and setting them equal to each other and solving.  Then we discovered we could use:

 

 (XX1)/3   and then subtract that from the relevant midpoint's X value to get the centroid's X.  Then the same for the Y value.  (which was a whole lot faster than setting two lines equal to each other and solving the equations, as we had been doing.)     e.g.   

 

MBCis 2.5 .   ( 2.5 - (-2))/3 = 3/2    

 

If I then subtract the -3/2 to the X value for the Midpoint of BC, I get 1 which if the correct value for the X coordinate of the centroid.  

 

 

But summing all 3 and dividing by 3 is way better!

 

 

And are there any other marvelously faster ways to get to the answers for the other types of centers? We are still working them via  getting an equation for each of two lines and setting them equal to each other and working it out?

 

 

 

 

 

 

Problem 1’s circumcenter took me about 30mins working out and double checking in my caffeine deficit mode. My night owl also makes mistakes when sleepy, he just woke up.

 

 

 

 

That's helpful to know.  I think I  need to help ds to be very careful  and to use the color and so on ideas to get each right if possible the first time.

 

Ds has around 50 homework problems to do this weekend.  Some are already done.  And some are going to be faster, but if what is just 1/3 of a problem will take 30 minutes or so in many cases, this is going to be a rough weekend.

 

Though I guess much better than doing them wrong over and over in 15 minutes each time.

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I assume the centroid formula would work for all triangles with vertices (real numbers) given. I’ll see if I can find any proof for that formula later as I’m at BK now having dinner. I got the formula from this link https://www.mathopenref.com/coordcentroid.html

 

I think subsequent problems can be solved faster due to familiarity of using the same steps repeatedly but geometry problems really can be more haste less speed. Still 50 problems seems plenty for homework.

 

ETA:

No “shortcut†for circumcenter and orthocenter. I find it easier to do the question slower if tired than to spot my mistakes after. Jumping jacks after each problem helps (for blood circulation) :)

Edited by Arcadia

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Off to the side, or for a nasty problem, on another sheet of paper, you should label every single variable with what it signifies, in words. Then, when you find out the numerical value, put it by the variable. e.g. Q = midpoint of AB. Then when you get tangled up with letters, you can look back to your reference sheet, and you don't have to scan through your paper looking for the numbers. 

 

AGREED! 

 

(And I would additionally stress that if you're making notes as part of solving a problem, they are part of your solution and as such should be included in the "showing your work" part of your solution.  That is, don't then erase your notes after solving the problem, or toss the extra page.  Lots of students do this all. the. time., and I am constantly reminding them that if you wrote something down to solve a problem, it's part of your work and needs to be shown, not erased or tossed!)

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