A right triangle has its right angle at (-4,1) and the equation of one of its legs is 2x-3y+11=0. Find the equation of the other leg
To find the equation of the other leg of the right triangle, we need to find its slope first.
The equation of the given leg is 2x - 3y + 11 = 0. To find the slope, we need to rewrite this equation in slope-intercept form (y = mx + b), where m is the slope.
Let's rearrange the given equation to solve for y:
2x - 3y + 11 = 0
-3y = -2x - 11
y = (2/3)x + 11/3
So, the slope of the given leg is 2/3.
Since the other leg of the right triangle is perpendicular to the given leg, its slope will be the negative reciprocal of 2/3.
The negative reciprocal of 2/3 is -3/2.
Now that we have the slope of the other leg (-3/2), and we know that the right angle of the triangle is at (-4, 1), we can use point-slope form to write the equation of the other leg.
Using point-slope form: y - y1 = m(x - x1), where (x1, y1) is the point (-4, 1) and m is the slope (-3/2), we can write:
y - 1 = (-3/2)(x - (-4))
y - 1 = (-3/2)(x + 4)
y - 1 = (-3/2)x - 6
y = (-3/2)x - 5
Therefore, the equation of the other leg of the right triangle is -3/2x - y - 5 = 0.
To find the equation of the other leg of the right triangle, we can use the fact that the slopes of the two legs of a right triangle are negative reciprocals of each other.
First, let's find the slope of the given leg: 2x - 3y + 11 = 0.
Rearrange the equation to slope-intercept form (y = mx + b):
2x - 3y + 11 = 0
-3y = -2x - 11
y = (2/3)x + 11/3
The slope of this leg is 2/3.
Since the slopes of the two legs of a right triangle are negative reciprocals of each other, the slope of the other leg will be the negative reciprocal of 2/3.
The negative reciprocal of 2/3 is -3/2.
Now we have the slope (-3/2) of the other leg. We also have the right angle at (-4, 1), which means the other leg should pass through that point.
Using the point-slope form of a linear equation, which is y - y1 = m(x - x1), we can substitute the values into the equation:
y - 1 = (-3/2)(x - (-4))
y - 1 = (-3/2)(x + 4)
y - 1 = (-3/2)x - 6
y = (-3/2)x - 5
Thus, the equation of the other leg of the right triangle is given by y = (-3/2)x - 5.
The legs are perpendicular. The slope of the given leg is 2/3.
So, the slope of the other leg is -3/2.
You now have a point and a slope, so the equation of the other leg is
y-1 = -3/2 (x+4)