The hypotenuse of a right angled triangle has its ends at the points and . Find an equation of the legs (perpendicular sides) of the triangle which are parallel to the axes.
The equations of the legs are: x = 1 and y = 1 OR x = -4 and y = 3.
Explanation
The question involves finding equations of legs of a right-angled triangle where the hypotenuse has endpoints at (1, 3) and (-4, 1), and the legs are parallel to the axes. Since one leg must be parallel to x-axis (horizontal) and the other parallel to y-axis (vertical), the right angle vertex must combine the x-coordinate from one endpoint and y-coordinate from the other endpoint. This gives two possible positions for the right angle vertex: (1, 1) or (-4, 3). For each position, we get corresponding equations of the legs.
Solution Steps
Step 1: Let the endpoints of hypotenuse be A(1, 3) and B(-4, 1).
Step 2: Since legs are parallel to axes, one leg is horizontal (parallel to x-axis) and other is vertical (parallel to y-axis).
Step 3: The right angle vertex C must have coordinates combining x from one endpoint and y from other endpoint.
Step 4: Case 1: C = (1, 1). Leg AC is vertical through x = 1, so equation is x = 1. Leg BC is horizontal through y = 1, so equation is y = 1.
Step 5: Case 2: C = (-4, 3). Leg AC is horizontal through y = 3, so equation is y = 3. Leg BC is vertical through x = -4, so equation is x = -4.