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

Mason Adams · Accepted Answer

Endpoints of altitude are #(2,1) and (5,2)#
length of altitude is #sqrt10#

To calculate d the distance between two points, use the distance formula :
#d=sqrt((x_2−x_1)^2+(y_2−y_1)^2)#, where #(x_1,y_1)# and #(x_2,y_2)# are the coordinates of the two points.

Given #A(1,4), B(2,1), and C(5,2)#,
# AC=sqrt((5-1)^2+(2-4)^2)=sqrt20, => AC^2=20#
#AB=sqrt((2-1)^2+(1-4)^2)=sqrt10, => AB^2=10#
#BC=sqrt((5-2)^2+(2-1)^2)=sqrt10, => BC^2=10#
As #AC^2=AB^2+BC^2, Delta ABC# is a right triangle, right-angled at #B#.

Hence, #BC# is the altitude perpendicular to #AB# from #C#
#=># endpoints of the altitude are #(2,1) and (5,2)#
length of the altitude #= BC = sqrt10#

Violet Aldrich · Answer

undefined To find the endpoint and length of the altitude going through corner C, first, calculate the slope of the line segment AB, then find the equation of the perpendicular line passing through C. The intersection point of this perpendicular line with the line segment AB will be the endpoint of the altitude. Finally, calculate the length of this altitude.

1. Calculate the slope of AB:
$$ m_{AB} = \frac{y_B - y_A}{x_B - x_A} $$

2. The slope of the perpendicular line will be the negative reciprocal of the slope of AB:
$$ m_{\perp} = -\frac{1}{m_{AB}} $$

3. Using point-slope form, find the equation of the perpendicular line passing through C:
$$ y - y_C = m_{\perp}(x - x_C) $$

4. Find the intersection point of this line with the line segment AB, which is the endpoint of the altitude.

5. Calculate the distance between this endpoint and point C to find the length of the altitude.

Let's proceed with the calculations:

1. Slope of AB:
$$ m_{AB} = \frac{1 - 4}{2 - 1} = -3 $$

2. Slope of the perpendicular line:
$$ m_{\perp} = -\frac{1}{-3} = \frac{1}{3} $$

3. Equation of the perpendicular line passing through C:
$$ y - 2 = \frac{1}{3}(x - 5) $$

Expanding:
$$ y - 2 = \frac{1}{3}x - \frac{5}{3} $$

$$ y = \frac{1}{3}x - \frac{5}{3} + 2 $$

$$ y = \frac{1}{3}x + \frac{1}{3} $$

4. Finding intersection point with AB:

$$ \frac{1}{3}x + \frac{1}{3} = -3x + 7 $$

$$ \frac{1}{3}x + 3x = 7 - \frac{1}{3} $$

$$ \frac{10}{3}x = \frac{20}{3} $$

$$ x = 2 $$

$$ y = -3(2) + 7 = 1 $$

The intersection point is (2, 1).

5. Calculate the length of the altitude:

$$ \text{Altitude length} = \sqrt{(5 - 2)^2 + (2 - 1)^2} $$

$$ \text{Altitude length} = \sqrt{3^2 + 1^2} $$

$$ \text{Altitude length} = \sqrt{9 + 1} $$

$$ \text{Altitude length} = \sqrt{10} $$

So, the endpoint of the altitude through corner C is (2, 1), and its length is $ \sqrt{10} $.