How do u find the perpenicular bisectors of a triangle and how to find the intersection of the the bisectors?

Answer 1

First off, a perpendicular bisector (by definition) is a line that cuts something in half.

We know that a triangle is composed of three lines and therefore (as I’m sure you’ve already presumed) has three perpendicular bisectors.

So lets try to find one. Its rather difficult without a triangle to show you on but we’ll make do with words.

First, find the midpoint of any given line of your triangle. You can do this by measuring it (if it’s to scale) or by using a midpoint formula (you can only do this if your triangle is on a graph). If measuring, find the length of the side and divide by two for the distance from the start of the line to the middle of it. If it is on a graph you can use the formula: ((x1+x2)/2),((y1+y2)/2). You’ve guessed it, this formula does the exact same thing as the first method but with numbers instead of with a ruler. In case you’re wondering; x1 refers to the x value of the first point of the line, y1 refers to the y value of the first point of the line, x2 refers to the x value of the second point of the line, and y2 refers to the y value of the second point of the line. You will end up with an answer that looks like: (x,y). This is the coordinate of your midpoint!

Now that you have the midpoint you must connect it to the opposite vertex of your triangle. This is easily done by hand and with a ruler. You’ll notice how this has divided the triangle into two exact halves. If you’re doing this in the mathematically sound way (i.e. with many formulas and calculations) please keep reading, your task is much more laborious! You need to find the slope between the midpoint (M) and your vertex (V). So we must use the slope formula: slope=(y2-y1)/(x2-x1). Where (x1,y1) are the coordinates of the midpoint and (x2,y2) are the coordinates of the vertex. Now that your slope is achieved you need to find the y-intercept. This is found by using the equation: y=mx+b. Sub in your midpoint coordinates for x and y (since they are a point on this line) and rearrange the equation so you have an answer for b (the y-intercept). Now that you have an equation for this bisector, draw it in and give yourself a pat on the back!

Repeat this process for the next two lines in your triangle. Here’s a hint though, you don't actually need to find the third bisector because it will just confirm what the last two have told you! Besides, you only need to have two bisectors to find the intersection of the bisectors.

Speaking of which, now we are going to find the intersection. If you’ve been doing it all by hand and ruler you simply just need to put a point where your bisectors have met and show that you have, indeed, found the intersection. If you have to do it the arduous method (in which case I applaud you), please read on. Since you have you have the equations for each bisector, this will be easy. You are simply going to find where these two lines intersect. If you have a Texas Instruments Graphing Calculator (or similar device), now would be the time to whip it out and graph these two lines. If you don’t have this device, have no fear; algebra is here!

If you have a pretty good understanding of algebra or even just pre-algebra you’ll understand this part fairly quickly. We have two equations and two unknowns. We have to solve for x and y in these two equations. This will give us the x-value and y-value that these two lines share. I’m going to use the two following equations to aid both my instruction and your understanding.

Lets say the two lines are y=2x-3 and y=x+1. We know that y=2x-3 in the first equation so we’ll sub it into the second equation. Therefore 2x-3=x+1. From here you need to move the xs to one side and the numbers to another, so: 2x-x=1+3 which become x=4 (don’t forget that the number’s sign (+ or -) is inversed when you change sides (change side, change sign). Now that we know x, we can sub x’s value into the first equation. Which becomes y=2(4)-3 and then y=5. So we know our intersection is (4,5).

I hope this has been an informative answer. It took some time to write but I think you and many others will find it useful. Good luck in your mathematical endevours!

Answer 2

Do you mean the traditional problem of finding the bisector using a compass and straightedge?

1) Measure off from a vertex an equal distance along its 2 lines

2) set your compass to the distance between the 2 points just marked (or any other sufficiently large distance)

3) using the compass draw a circle around both of the 2 points marked off in step 1

4) use the straightedge draw the line from the vertex through the intersections of the 2 circles. This is the perpendicular bisector.