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

Sophia Alfred · Accepted Answer

The end points are #=(71/17,131/17)# and the length is #=1.37#

The triangle's corners are

#A=(7,3)#

#B=(4,8)#

#C=(3,7)#

The slope of the line #AB# is #m=(8-3)/(4-7)=-5/3#

The equation of line #AB# is

#y-8=-5/3(x-4)#

#y-8=-5/3x+20/3#

#y+5/3x=44/3#...........................#(1)#

#mm'=-1#

The slope of the line perpendicular to #AB# is #m'=3/5#

The equation of the altitude through #C# is

#y-7=3/5(x-3)#

#y-7=3/5x-9/5#

#y=3/5x+26/5#................................#(2)#

Solving for #x# and #y# in equations #(1)# and #(2)#, we get

#-5/3x+44/3=3/5x+26/5#

#3/5x+5/3x=44/3-26/5#

#34/15x=132/15#

#x=142/34=71/17#

#y=3/5*71/17+26/5=655/85=131/17#

The end points of the altitude is #=(71/17,131/17)#

The altitude's duration is

#=sqrt((3-71/17)^2+(7-131/17)^2)#

#=sqrt((-20/17)^2+(-12/17)^2)#

#=sqrt(544)/17#

#=1.37#

Allison Allen · Answer

End points are #(3,7) and (4.18, 7.71)#
Length of altitude is # 1.38# unit

Let #CD# be the altitude going through #C# touches #D# on line #AB#. #C# and #D# are the endpoints of altitude #CD. CD# is perpendicular on #AB#. Slope of #AB= m_1= (y_2-y_1)/(x_2-x_1)=(8-3)/(4-7) = -5/3 :. # Slope of #CD=m_2= -1/m_1= 3/5 # Equation of line #AB# is # y - y_1 = m_1(x-x_1) or y- 3 = -5/3(x-7)# or #3y-9 = -5x + 35 or 5x+3y= 35+9 or 5x+3y = 44 (1)# Equation of line #CD# is # y - y_3 = m_2(x-x_3) or y- 7 = 3/5(x-3)# or #5y-35 = 3x -9 or 3x-5y= -35+9 or 3x-5y = -26 (2)# By solving equations (1) and (2), we can determine D's coordinates. #15x+9y = 132 (3)# # 15x - 25y = -130 (4)# Subtracting we get, #34y=262 or y =131/17=7.71 ; x = (44-3*131/17)/5= (44-393/17)/5 =(748-393)/85 =355/85=4.18# So end points are #(3,7) and (4.18, 7.71)# Length= # sqrt((3-4.18)^2+ (7-7.71)^2)) 1.38 # unit [Ans]

Savannah Aguilar · Answer

undefined To find the altitude going through corner C of the triangle, we first need to determine the equation of the line containing side AB, which will be perpendicular to the altitude passing through corner C. Then, we can find the intersection point of this line with the altitude to determine its endpoints. Finally, we calculate the distance between the intersection point and corner C to find the length of the altitude.

1. Calculate the slope of the line passing through corners A and B using the formula $ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} $.
$$ m_{AB} = \frac{{8 - 3}}{{4 - 7}} = \frac{5}{-3} = -\frac{5}{3} $$

2. Since the altitude through corner C is perpendicular to side AB, the slope of the altitude is the negative reciprocal of the slope of side AB. Therefore, the slope of the altitude ($ m_{\text{altitude}} $) is the negative reciprocal of $ m_{AB} $, which is $ \frac{3}{5} $.

3. Using the point-slope form of the equation of a line $ y - y_1 = m(x - x_1) $, where $ (x_1, y_1) $ is a point on the line, we can write the equation of the altitude passing through corner C using point C ($ (3, 7) $) and the slope $ \frac{3}{5} $:
$$ y - 7 = \frac{3}{5}(x - 3) $$

4. Solve this equation for $ y $ to find the equation of the altitude:
$$ y = \frac{3}{5}x + 5 $$

5. Next, we find the intersection point of this line with the line passing through corner C, which is the altitude. We set the equations of the altitude and the line passing through corner C equal to each other and solve for $ x $ and $ y $.
$$ \frac{3}{5}x + 5 = 7 $$
$$ \frac{3}{5}x = 2 $$
$$ x = \frac{10}{3} $$

6. Substitute $ x = \frac{10}{3} $ into either equation to find $ y $:
$$ y = \frac{3}{5} \left(\frac{10}{3}\right) + 5 = \frac{18}{5} + 5 = \frac{18}{5} + \frac{25}{5} = \frac{43}{5} $$

7. So, the intersection point of the altitude with corner C is $ \left(\frac{10}{3}, \frac{43}{5}\right) $.

8. Finally, calculate the length of the altitude, which is the distance between this intersection point and corner C using the distance formula:
$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
$$ d = \sqrt{\left(\frac{10}{3} - 3\right)^2 + \left(\frac{43}{5} - 7\right)^2} $$
$$ d = \sqrt{\left(\frac{1}{3}\right)^2 + \left(\frac{3}{5}\right)^2} $$
$$ d = \sqrt{\frac{1}{9} + \frac{9}{25}} $$
$$ d = \sqrt{\frac{25 + 81}{225}} $$
$$ d = \sqrt{\frac{106}{225}} $$
$$ d = \frac{\sqrt{106}}{15} $$

So, the length of the altitude going through corner C is $ \frac{\sqrt{106}}{15} $, and its endpoints are $ \left(\frac{10}{3}, \frac{43}{5}\right) $ and $ (3, 7) $.