"We require to find the equations of 2 altitudes and "We require to find the equations of 2 altitudes and
"solve them simultaneously for orthocentre"solve them simultaneously for orthocentre
"label the vertices"label the vertices
A=(2,2),B=(5,1)" and "C=(4,6)A=(2,2),B=(5,1) and C=(4,6)
color(blue)"Altitude from vertex C to AB"Altitude from vertex C to AB
"calculate slope m using "color(blue)"gradient formula"calculate slope m using gradient formula
•color(white)(x)m=(y_2-y_1)/(x_2-x_1)∙xm=y2−y1x2−x1
m_(AB)=(1-2)/(5-2)=-1/3mAB=1−25−2=−13
m_("altitude")=-1/m=-1/(-1/3)=3maltitude=−1m=−1−13=3
"using "m=3" and "(a,b)=(4,6)using m=3 and (a,b)=(4,6)
y-6=3(x-2)larry-b=m(x-a)y−6=3(x−2)←y−b=m(x−a)
y-6=3x-6y−6=3x−6
y=3xto(1)y=3x→(1)
color(blue)"Altitude from vertex A to BC"Altitude from vertex A to BC
m_(BC)=(6-1)/(4-5)=-5mBC=6−14−5=−5
m_("altitude")=-1/(-5)=1/5maltitude=−1−5=15
"using "m=1/5" and "(a,b)=(2,2)using m=15 and (a,b)=(2,2)
y-2=1/5(x-2)y−2=15(x−2)
y-2=1/5x-2/5larrcolor(blue)"multiply through by 5"y−2=15x−25←multiply through by 5
5y-10=x-25y−10=x−2
5y=x+85y=x+8
y=1/5x+8/5to(2)y=15x+85→(2)
"solving equations "(1)" and "(2)solving equations (1) and (2)
3x=1/5x+8/5rArrx=4/73x=15x+85⇒x=47
y=3xx4/7=12/7y=3×47=127
"orthocentre "=(4/7,12/7)orthocentre =(47,127)
