Is $△ O H J$ 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, $\overline{H M}$ 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 $\overline{O M}$ and $\overline{H M}$

To determine the length of the last side, we have to use the Distance Formula.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstitutePoints

Substitute $(m, n)$ & $(0, 0)$

d_{\overline{O H}} = (m - 0)^{2} + (n - 0)^{2}

SubTerm

Subtract term

d_{\overline{O H}} = m^{2} + n^{2}

Let's summarize the side lengths:

O M = m, H M = n, O H = 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	$M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$	Midpoint
$\overline{O M}$	$(m, 0), (0, 0)$	$M (\frac{m + 0}{2}, \frac{0 + 0}{2})$	$M (\frac{m}{2}, 0)$
$\overline{H M}$	$(m, n), (m, 0)$	$M (\frac{m + m}{2}, \frac{n + 0}{2})$	$M (m, \frac{n}{2})$
$\overline{O H}$	$(m, n), (0, 0)$	$M (\frac{m + 0}{2}, \frac{n + 0}{2})$	$M (\frac{m}{2}, \frac{n}{2})$

Let's summarize the midpoints:

M_{\overline{O M}} (\frac{m}{2}, 0), M_{\overline{H M}} (m, \frac{n}{2}), M_{\overline{O H}} (\frac{m}{2}, \frac{n}{2})

Is △OHJ a right triangle?

Side lengths

Midpoints

Is $△ O H J$ a right triangle?

Exercise