finding equations of trangle altitudes.

[allegansveritatem]

Here is the exercise:


Here is how I found the dimensions of the sides and the equations for the sides:


And here is how I found the altitudes using Herons formula and using the resultant area with the formula A=1/2 bh to find the dimensions of the altitudes:


I don't know if this is the best way to go about getting this information but that is how I did it and with all this accomplished I still have no idea how to get equations for the altitudes when I only know one point and the length for each altitude. And then there is the circumstance that this exercise is meant to be solved with some kind of use of the systems of equations with two variables technique. Where to go from here?

 

[joshuaa]

bravo. i can see that the first step you did was finding the slopes

you will not believe me if i told you that this step was enough to find the equations of the altitudes

for example, you found this slope, [MATH]\frac{1}{4}[/MATH]
then the first equation you need is perpendicular to that slope (it has a slope of [MATH]-4[/MATH])

this means, the first equation is

[MATH]y - y_0 = -4(x - x_0)[/MATH]
can you see which vertex point this line passes?

 

[allegansveritatem]

bravo. i can see that the first step you did was finding the slopes

you will not believe me if i told you that this step was enough to find the equations of the altitudes

for example, you found this slope, [MATH]\frac{1}{4}[/MATH]
then the first equation you need is perpendicular to that slope (it has a slope of [MATH]-4[/MATH])

this means, the first equation is

[MATH]y - y_0 = -4(x - x_0)[/MATH]
can you see which vertex point this line passes?

I will have to think about what you are saying here. It is late now and my noodle is too soft to work with right now.. I begin to have a vague memory of the fact that one line perpendicular to another has a slope that is the reciprocal of the one it meets...I will come back to this tomorrow and later in the day I will post my results. Thanks for the pointer.

 

[allegansveritatem]

bravo. i can see that the first step you did was finding the slopes

you will not believe me if i told you that this step was enough to find the equations of the altitudes

for example, you found this slope, [MATH]\frac{1}{4}[/MATH]
then the first equation you need is perpendicular to that slope (it has a slope of [MATH]-4[/MATH])

this means, the first equation is

[MATH]y - y_0 = -4(x - x_0)[/MATH]
can you see which vertex point this line passes?

I went at it again today with your tip re: perpendicular lines in mind and their slopes and came up with this:
First I worked out the slopes of the sides of the triangle thus:

Then I the reciprocals of the slopes (negative in the case of positive and positive in the case of negative) and used the point slope formula to get equations:


after that I cheated and put created a graph of these lines to see if they intersected. They did and so now I want to go back to this tomorrow and work out the intersection point algebraically which was the point of the whole exercise from the get go.But here is an image of the graph:

 

[pka]

after that I cheated and put created a graph of these lines to see if they intersected. They did and so now I want to go back to this tomorrow and work out the intersection point algebraically which was the point of the whole exercise from the get go.But here is an image of the graph:
View attachment 21634

Why in the world would you say that it was cheating to use technology to find the intersection?
I have tried to get students to use such programs. We even have a site licence for WolframAlphaPro.
But very few will take advantage. Even that does not change my prediction that future calculus textbooks
will not contain a chapter on techniques of integration.