A triangle has corners A, B, and C located at #(1 ,8 )#, #(2 ,3 )#, and #(5 ,9 )#, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

#21/\sqrt26# & #(235/26, 255/26), (25/26, 213/26)#

The vertices of #\Delta ABC# are #A(1, 8)#, #B(2, 3)# & #C(5, 9)#
The area #\Delta# of #\Delta ABC# is given by following formula
#\Delta=1/2|1(3-9)+2(9-8)+5(8-3)|#
#=21/2#
Now, the length of side #AB# is given as
#AB=\sqrt{(1-2)^2+(8-3)^2}=\sqrt26#
If #CN# is the altitude drawn from vertex C to the side AB then the area of #\Delta ABC# is given as
#\Delta =1/2(CN)(AB)#
#21/2=1/2(CN)(\sqrt26)#
#CN=21/\sqrt26#
Let #N(a, b)# be the foot of altitude CN drawn from vertex #C(5, 9)# to the side AB then side #AB# & altitude #CN# will be normal to each other i.e. the product of slopes of AB & CN must be #-1# as follows
#\frac{b-9}{a-5}\times \frac{3-8}{2-1}=-1#
#a=5b-40\ ............(1)#

Now, the length of altitude CN is given by distance formula

#\sqrt{(a-5)^2+(b-9)^2}=21/\sqrt26#
#(5b-40-5)^2+(b-9)^2=(21/\sqrt26)^2#
#(b-9)^2=21^2/26^2#
#b=255/26, 213/26#
Setting above values of #b# in (1), we get the corresponding values of #a#
#a=235/26, 25/26#

hence, the end points of altitudes are

#(235/26, 255/26), (25/26, 213/26)#
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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 passing through points A and B. Let's call this slope ( m_{AB} ).
  2. The perpendicular slope to ( m_{AB} ) is the negative reciprocal of ( m_{AB} ), denoted as ( -\frac{1}{m_{AB}} ). Let's call this ( m_{\perp} ).
  3. Using point C and the slope ( m_{\perp} ), find the equation of the line passing through C that is perpendicular to AB.
  4. Find the intersection point of this perpendicular line and the line containing AB. This point will be the foot of the altitude, let's call it D.
  5. The distance from point C to point D is the length of the altitude.

Once you find the endpoint D, you will have the altitude and its length.

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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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