What is the orthocenter of the triangle formed by the intersection of the lines #x = 4# , #y = 1/2x + 7# , and# y = -x + 1#?

1 Answer
Mar 22, 2016

Answer:

The Orthocenter, O is#=>O(0,5)#

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Explanation:

The orthocenter is the point where all three altitudes of the triangle intersect. An altitude is a line which passes through a vertex of the triangle and is perpendicular to the opposite side. So we need to find lines that are perpendicular to the give equation and where at least 2 of them intersect:

A perpendicular line to any line, #y=mx+b# is give by:
#y_(per) = m_(per)x+ b_(per); m_(per)= -1/m# =====> (1)
First we need to get the coordinates of the vertices,
Let's solve equations:
Given #y_1=1/2x+7# and #x=4# simultaneously to get #y=11#
#y_2=-x+1 # and #x=4# simultaneously to get #y=3#
#y=1/2x+7, y=-x+1; 1/2x+7=-x6+1; x=-4; y = 5#
Thus the vertices of the triangle are given by:
#A(-4,5), B(4,11)# and #C(4,-3)#

Now let find the perpendicular line to #y_1# that passes through point #C(4,-3)# from (1) #m_(per)= -1/(1/2)= -2#
#y_(per_1)= -2x+ b_(per)# let #x=4; y = -3 " and solve for " b_(per)#
#-3=-2(4)+ b_(per); b_(per)=5#
So what we have is #y_(per_1) = -2x+5#
and the line perpendicular to x=4 passing through A(-4,5) is #y=5#

Thus the Orthocenter is #y_(per_1) = -2x+5 = 5# Solve for #x#
#x= 0# #:. "the Orthocenter," =>O(0,5)#