Question

Triangle PQR has vertices P(4,-1),Q(-2,7), and R(9,9). Find an equation of the median and altitude from R.

Answer 1

Both of the equation of the median and altitude from R are the same: y=(3/4)x + 9/4, this means that they are at equal length so the triangle PQR is an isosceles (a triangle that has two sides of the same length).

Answer 2

If we are drawing altitude from R the PQ will be base of this triangle. So we can say Median from R will cut on Midpoint of PQ. So firstly we will find mid point of PQ Which is S given as below:
P(4,-1) , Q (-2,7)

Let Co-ordinate of S is (x,y)
So
`x = (x_1+x_2)/(2)`

`x = (4-2)/(2) = (2)/(2) = 1`


`y = (y_1+y_2)/(2)`

`y = (-1+7)/(2) = (6)/(2)= 3`
So midpoint of PQ is S(1,3).

Now we can find equation of RS
Co-ordinate of R is (9,9) and and S is (1,3).

In general equation of line from two point
`(x_1 , y_1)` and
`(x_2,y_2)` is

`(y-y_1) = (y_2-y_1)/(x__2-x_1)(x-x_1)`

So equation of RS is


`(y-9) = (9-3)/(9-1) (x-1)`

`(y-9) = (6)/(8)(x-1)`

`8(y-9) = 6(x-1)`

`8y-72 = 6x - 6`

`-72+6 = 6x - 8y`

`-66 = 6x-8y`

So the equation of median is 6x - 8y = -66