Q.The hypotenuse of a right angled triangle has its ends at the points $(1, 3)$ and $(-4, 1)$. Find an equation of the legs (perpendicular sides) of the triangle which are parallel to the axes.

Concept: Equations of lines parallel to the coordinate axes.

Reasoning:

1. The hypotenuse joins $(1,3)$ and $(-4,1)$. The legs are parallel to the axes, so one leg is vertical ($x = \text{constant}$) and the other is horizontal ($y = \text{constant}$).

2. The right angle must be at the intersection of a vertical line through one endpoint and a horizontal line through the other endpoint.

3. Two possibilities exist:

   • Vertical through $(1,3)$: $x = 1$; horizontal through $(-4,1)$: $y = 1$. Intersection at $(1,1)$. …

We have a right-angled triangle whose hypotenuse joins $(1, 3)$ and $(-4, 1)$. The legs (the two perpendicular sides) are parallel to the axes. That means one leg is horizontal (parallel to the x-axis) and the other is vertical (parallel to the y-axis). The right angle is therefore at the intersection of a horizontal line through one endpoint and a vertical line through the other endpoint.

Let’s see why this is the only possibility. If a leg is parallel to the x-axis, its equation is $y = \text{constant}$. If the other leg is parallel to the y-axis, its equation is $x = \text{constant}$. The right angle is at the point where these two lines meet. That point must also be the third vertex of the triangle, distinct from the two given endpoints of the hypotenuse.

So the third vertex (the right angle) has coordinates $(h, k)$ where $h$ is the x-coordinate of one endpoint and $k$ is the y-coordinate of the other endpoint. There are two natural ways to pair them.

1. First possibility: Take the x-coordinate from $(1, 3)$ and the y-coordinate from $(-4, 1)$. That gives the point $(1, 1)$.

   • The vertical leg through $(1, 1)$ and $(1, 3)$ is the line $x = 1$.
   • The horizontal leg through $(1, 1)$ and $(-4, 1)$ is the line $y = 1$. These two legs are perpendicular (one vertical, one horizontal), and the hypotenuse joins $(1, 3)$ and $(-4, 1)$. So the triangle is right-angled at $(1, 1)$.

2. Second possibility: Take the x-coordinate from $(-4, 1)$ and the y-coordinate from $(1, 3)$. That gives the point $(-4, 3)$.

   • The vertical leg through $(-4, 3)$ and $(-4, 1)$ is the line $x = -4$. …