The only point where we know both coordinates is A(0,0). Let's start by plotting this point.
Additionally, we know that C(2h, 0) is on the x-axis because it's y-coordinate is 0. Let's mark this point as well. Since both A and C have the same y-coordinate, the side between these points will be horizontal.
As for the third point, B(h,h), we know it's x-coordinate is in the middle of AC which means â–ł ABC is an isosceles triangle. Additionally, the y-coordinate is h units above the x-axis.
Finding length and slope
To find the slope of the sides, we use the Slope Formula. Note that horizontal lines have a slope of 0 so we do not have to determine the slope of AC.
Side
Points
y_2-y_1/x_2-x_1
m
AB
( h,h), ( 0,0)
h- 0/h- 0
1
BC
( h,h), ( 2h,0)
h- 0/h- 2h
- 1
To find the length of the sides, we can use the Distance Formula. However, we do not need to use the formula for the horizontal side as it's the absolute value of the difference between the endpoints' x-coordinates.
AC: |2h-0|=2h
The remaining sides we have to calculate with the Distance Formula.
Distance
Points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
d
AB
( h,h), ( 0,0)
sqrt(( h- 0)^2+( h- 0)^2)
hsqrt(2)
BC
( 2h,0), ( h,h)
sqrt(( 2h- h)^2+( 0- h)^2)
hsqrt(2)
Let's summarize the length and slope:
ll
Slope & Distance
m_(AC)=0 & d_(AC)=2h
m_(AB)=1 & d_(AB)=hsqrt(2)
m_(BC)=- 1 & d_(BC)=hsqrt(2)
Midpoints
To find the midpoint of any side we can use the Midpoint Formula.
Side
Points
M(x_1+x_2/2,y_1+y_2/2)
Midpoint
AC
( 2h,0), ( 0,0)
M(2h+ 0/2,0+ 0/2)
M(h,0)
AB
( h,h), ( 0,0)
M(h+ 0/2,h+ 0/2)
M(h/2,h/2)
BC
( 2h,0), ( h,h)
M(2h+ h/2,0+ h/2)
M(3h/2,h/2)
Let's summarize the midpoints:
M_(AC)(h,0), M_(AB)(h/2,h/2), M_(BC)(3h/2,h/2)
