Find the coordinates of the orthocenter of the triangle formed by straight lines?

#x-y-5=0# , #2x-y-8=0# and #3x-y-9=0#

#x-y-5=0# , #2x-y-8=0# and #3x-y-9=0#

1 Answer
Write your answer here...
Start with a one sentence answer
Then teach the underlying concepts
Don't copy without citing sources
preview
?

Answer

Write a one sentence answer...

Answer:

Explanation

Explain in detail...

Explanation:

I want someone to double check my answer

Describe your changes (optional) 200

2
May 2, 2017

Answer:

The orthocentre is at #(–6, 1)#.

Explanation:

To find the orthocentre of a triangle, we connect each vertex to its opposite side with a perpendicular line. All three of these lines will intersect at a single point, called the orthocentre.

upload.wikimedia.org

(Note that for obtuse triangles, the orthocentre will be outside the triangle.)

Let the triangle's vertices be #A#, #B#, and #C#. Our process will involve 3 steps:

  1. Find the equation of a line that passes through #C#, and is perpendicular to #bar(AB)#.
  2. Find the equation of a line that passes through #B#, and is perpendicular to #bar(AC)#.
  3. Solve the system of linear equations made from the equations in steps 1 and 2.

(We don't need to bother with finding the third line, since it will be redundant.)

Uh-oh—looks like we don't know the coordinates of the vertices. We'll have to find two of them first.

Let's call the line given by the first equation "side #a#", and the middle equation "side #b#". Solving this system of two equations gives

#color(white)(=>)y=x-5=2x-8#
#=>color(white)(y=x-)3=x#

#=>y=3-5#
#color(white)(=>y)=–2#

Thus, #a# and #b# intersect at #C=(3, –2)#.

The last equation gives us side #c# (also called #bar(AB)#). We now find the line that passes through #(3, –2)# and is perpendicular to #3x-y-9=0#.

Perpendicular lines have slopes that are negative reciprocals of each other. (Here's an explanation why: https://socratic.org/featured?s=365506) The slope of #3x-y-9=0# is 3 (all we do is move the #y# to the right, and the equation is in #y=mx+b# form). Thus, the slope of a perpendicular line to this one is #-1/3#.

We now want to find the equation of a line passing through #(x, y)=(3, –2)# with a slope of #m=–1/3#. Plug these into the equation of a line to find the necessary #y#-intercept:

#"   "y="    "mx"  "+b#
#–2=–1/3(3)+b#
#–2=" "-1"   "+b#
#–1=b#

So the equation of the altitude from #bar(AB)# to #C# is #y=–1/3x-1#. Step 1 is done!

For brevity, I'll leave step 2 to you as an exercise. The equation is #y=–1/2x-2#.

Now, all we do is find where these two lines meet (step 3). Set the equations equal to each other and solve:

#y=–1/3x-1=–1/2x-2#

#color(white)(y=)"  "–2x-6=–3x-12# (multiply through by 6)

#"                      "x=–6#

#=>y=–1/3(–6)-1#

#color(white)(=>)y=2-1#
#color(white)(=>)y=1#

There we go! After all that, the orthocentre of the triangle is at #(x, y)=(–6, 1)#.

Was this helpful? Let the contributor know!
1500
Impact of this question
1667 views around the world
You can reuse this answer
Creative Commons License