What do point O and point J have in common? How about H and J?
Right triangle?: Yes Side lengths: OM=m, HM=n, OH=sqrt(m^2+n^2) Midpoints: M_(OM)( m2,0), M_(HM)(m, n2), M_(OH)( m2, n2)
Practice makes perfect
Is â–ł OHJ a right triangle?
The only point where we know both coordinates is O(0,0). Let's start by plotting this point.
Additionally, we know that J(m, 0) is on the x-axis because it's y-coordinate is 0. Let's mark this point as well. Since both O and M are on the same y-coordinate, the side between these points will be horizontal.
As for the third point, H( m,n), we know it's on the same x-coordinate as J( m,n). Therefore, HM will be a vertical side. Since the triangle has one vertical side and one horizontal side, this is in fact a right triangle.
Side lengths
To find the length of any side in a coordinate plane, we can use the Distance Formula. However, for vertical or horizontal sides, the formula is not necessary when calculating the length:
Horizontal segments: The length of a vertical segment is the absolute value of the difference between the endpoints' x-coordinates.
Vertical segments: The length of a vertical segment is the absolute value of the difference between the endpoints' y-coordinates.
Using this, we can find OM and HM
To determine the length of the last side, we have to use the Distance Formula.
Let's summarize the side lengths:
OM=m, HM=n, OH=sqrt(m^2+n^2)
Midpoints
By the definition of a midpoint, we know that such a point bisects each of the three sides. 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
OM
( m,0), ( 0,0)
M(m+ 0/2,0+ 0/2)
M(m/2,0)
HM
( m,n), ( m,0)
M(m+ m/2,n+ 0/2)
M(m,n/2)
OH
( m,n), ( 0,0)
M(m+ 0/2,n+ 0/2)
M(m/2,n/2)
Let's summarize the midpoints:
M_(OM)(m/2,0), M_(HM)(m,n/2), M_(OH)(m/2,n/2)
