A triangle has corners A, B, and C located at #(6 ,4 )#, #(2 ,5 )#, and #(3 , 1 )#, respectively. What are the endpoints and length of the altitude going through corner C?

Question

A triangle has corners A, B, and C located at #(6   ,4   )#, #(2  ,5    )#, and #(3 ,  1   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Violet Aldrich · Accepted Answer

End point of altitude is at #(66/17,77/17)#
The altitude has a length of (approximately) #3.638#

The line segment #AB# with #A=(6,4)# and #B=(2,5)#
has a slope of #m_(AB)=(5-4)/(2-6)=-1/4#

Using the point #A# and this altitude we can develop an equation for this line using the slope-point form:
#color(white)("XXX")y-4=(-1/4)(x-6)#

#color(white)("XXX")4y-16=6-x#

#color(white)("XXX")x+4y= 22#

#color(teal)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")#

The altitude through #C=(3,1)# is perpendicular to the base #AB#
and therefore the altitude has a slope of #m_("alt")=-1/(m_(AB))=4#

Using the point #C# and the slope of the altitude we can develop an equation for the altitude using the slope-point form:
#color(white)("XXX")y-1=4(x-3)#

#color(white)("XXX")y-1=4x-12#

#color(white)("XXX")4x-y=11#

#color(teal)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")#

The point where the altitude intersects the base can be found as the solution to the set of equations:
[1]#color(white)("XXX")x+4y=22#
[2]#color(white)("XXX")4x-y=11#

Adding [1] and #4# times [2]
#color(white)("XXXXXX")x+4y=22#
#color(white)("XXXXX")underline(16x-4y=44)#
[3]#color(white)("XXXX")17xcolor(white)("XXX")=66#
or (after dividing by #17#)
[4]#color(white)("XXX")x=66/17#

Similarly, multiplying [1] by #4# and subtracting [2]
#color(white)("XXXXXX")(4x+16y)=color(white)("Xx")(88)#
#color(white)("XXXxX")underline(-(4x-color(white)("Xx")y)=-(11))#
[5]#color(white)("XXXXXXXxX")17y=color(white)("XxX")77#
or (after dividing by #17#)
[6]#color(white)("XXX")y=77/17#

#color(teal)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")#

To find the length of the altitude use the Pythagorean Theorem
to calculate the distance from #(3,1)# to #(66/17,77/17)#

#color(white)("XXX")"Alt." = sqrt((3-66/17)^2+(1-77/17)^2)#

Using a calculator, this approximates as
#color(white)("XXX")"Alt."~~3.638#

Sophia Alfred · Answer

undefined To find the endpoints of the altitude going through corner C, we first need to find the equation of the line containing side AB. Then, we determine the perpendicular line passing through point C. The intersection of these two lines will give us the endpoint of the altitude.

Step 1: Finding the equation of line AB:
$$ \text{Slope of AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 4}{2 - 6} = \frac{1}{-4} = -\frac{1}{4} $$
Using the point-slope form of a line, with point A (6, 4):
$$ y - y_1 = m(x - x_1) $$
$$ y - 4 = -\frac{1}{4}(x - 6) $$
$$ y - 4 = -\frac{1}{4}x + \frac{3}{2} $$
$$ y = -\frac{1}{4}x + \frac{11}{2} $$

Step 2: Finding the equation of the altitude through point C:
Since the altitude is perpendicular to side AB, its slope is the negative reciprocal of the slope of AB.
$$ \text{Slope of altitude} = \frac{1}{4} $$
Using the point-slope form of a line, with point C (3, 1):
$$ y - y_1 = m(x - x_1) $$
$$ y - 1 = \frac{1}{4}(x - 3) $$
$$ y - 1 = \frac{1}{4}x - \frac{3}{4} $$
$$ y = \frac{1}{4}x + \frac{1}{4} $$

Step 3: Finding the intersection point of the lines:
Solving the equations of the lines AB and the altitude will give us the coordinates of the endpoint of the altitude.
$$ -\frac{1}{4}x + \frac{11}{2} = \frac{1}{4}x + \frac{1}{4} $$
$$ -\frac{1}{4}x - \frac{1}{4}x = \frac{1}{4} - \frac{11}{2} $$
$$ -\frac{1}{2}x = \frac{1}{4} - \frac{22}{4} $$
$$ -\frac{1}{2}x = -\frac{21}{4} $$
$$ x = \frac{-21}{4} \times \frac{-2}{1} $$
$$ x = \frac{21}{2} $$
$$ y = \frac{1}{4} \times \frac{21}{2} + \frac{1}{4} $$
$$ y = \frac{21}{8} + \frac{2}{8} $$
$$ y = \frac{23}{8} $$
So, the endpoint of the altitude is (10.5, 2.875).

Step 4: Finding the length of the altitude:
We can use the distance formula to find the length of the altitude, which is the distance between point C and the endpoint we found.
$$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
$$ \text{Distance} = \sqrt{(10.5 - 3)^2 + (2.875 - 1)^2} $$
$$ \text{Distance} = \sqrt{(7.5)^2 + (1.875)^2} $$
$$ \text{Distance} = \sqrt{56.25 + 3.515625} $$
$$ \text{Distance} = \sqrt{59.765625} $$
$$ \text{Distance} \approx 7.73 $$

Therefore, the endpoints of the altitude through corner C are approximately (10.5, 2.875), and its length is approximately 7.73 units.