1. The first question is the equation of the line from point R as a median. It is a median if it intersects at the midpoint of the opposite length. This is represented as the red line. The midpoint is:
x,m = (4+-2)/2 = 1
y,m = (-1+7)/2 = 3
The midpoint is at (1,3). With this point and point R(9,9), the equation would be:
y = mx + b, where
m = (9 - 3)/(9 - 1) = 0.75
b is the y-intercept
Substituting any point,
3 = 0.75(1)+b
b = 2.25
Thus, the equation for the median is:
y = 0.75x + 2.25
2.) The altitude of the triangle from point r is a straight line from vertex R down to the opposite side which creates a right angle. As you can see in the picture, this is also the median. So, the equation is also y = 0.75x + 2.25.
3.) Yes, the answers for the median and altitude are the same, because the median also makes a perpendicular angle to the opposite length, which makes it the altitude.
4.) If the triangle is isosceles, then the length of sides QR and PR should be equal. Let's use the distance formula:
Between Q(-2,7) and R(9,9)
d = √(9 - ⁻2)² + (9 - 7)² = 5√5
Between P(4,-1) and R(9,9)
d = √(9 - 4)² + (9 - ⁻1)² = 5√5
Since the distances are equal, then the triangle is an isosceles.
