A triangle has corners A, B, and C located at #(8 ,3 )#, #(4 ,5 )#, and #(6 , 7 )#, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

Endpoints of Altitude: #(6,7)# and #(24/5,23/5)#

Length of Altitude: #~~2.68#

Given:
#color(white)("XXX")A=(8,3)#
#color(white)("XXX")B=(4,5)#
#color(white)("XXX")C=(6,7)#

First let's consider the line through #A# and #B#
#color(white)("XXX")"slope"_(AB)=(5-3)/(4-8)=-1/2#

#color(white)("XXX")#using the point-slope form the equation of the line through #A# and #B# is
#color(white)("XXXXXX")(y-3)=-1/2(x-8)#

#color(white)("XXXXXX")2y-6 =8-x#

#color(white)("XXXXXX")x+2y=14#

Looking at the altitude from #AB# through #C#
#color(white)("XXX")#The altitude line is always perpendicular to the base.
#color(white)("XXX")#Therefore
#color(white)("XXX")"slope"_"Alt"= 2# ...since the base has a slope of #(-1/2)#

#color(white)("XXX")#using the slope-point form the equation of the altitude line through #C# is
#color(white)("XXXXXX")y-7=2(x-6)#

#color(white)("XXXXXX")y-7=2x-12#

#color(white)("XXXXXX")2x-y=5#

Finding the point of intersection of the altitude with the base #AB#
#color(white)("XXX")#We have
#color(white)("XXXXXX"){(x+2y=14),(2x-y=5):}#
#color(white)("XXX")#Using standard solution techniques:
#color(white)("XXXXXX")(x,y)=(24/5,23/5)#

Calculating the length of the Altitude
#color(white)("XXX")#Using the Pythagorean Theorem
#color(white)("XXX")#with endpoints #(6,7)# and #(24/5,23/5)#

#color(white)("XXXXXX")abs("alt")=sqrt((6-24/5)^2+(7-23/5)^2)#

#color(white)("XXXXXXXX")~~2.68# (using calculator/spreadsheet)

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Answer 2

To find the altitude going through corner C of the triangle, follow these steps:

  1. Calculate the slope of the line containing side AB, which passes through corners A and B.
  2. Determine the negative reciprocal of the slope of side AB to find the slope of the altitude passing through corner C.
  3. Use the point-slope form to write the equation of the altitude passing through corner C using the coordinates of corner C and the slope of the altitude.
  4. Find the intersection point of this altitude with the line containing side AB to determine the endpoints of the altitude.
  5. Calculate the distance between the intersection point and corner C to find the length of the altitude.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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