The vertices of trianglePQR are P(1,3), Q(5,4) & R(5,15). Find the length of PR, area of TrianglePQR & length of per. line drown fm Q to PR.

To find the length of PR, we'll use the distance
formula:

PR= sqrt[(xR-xP)^2 +
(yR-yP)^2]

PR =
sqrt[(5-1)^2+(15-3)^2]

PR =
sqrt(16+144)

PR = sqrt 160

PR
= sqrt 16*10

PR = 4*sqrt
10

 Let's calculate the length of the segment
QP

QP =
SQRT[(xP-xQ)^2+(yP-yQ)^2]

QP =
sqrt[(1-5)^2+(3-4)^2]

QP =
sqrt(16+1)

QP = sqrt 17

Now,
let's calculate QR

QR = sqrt
[(5-5)^2+(15-4)^2]

QR = sqrt
11^2

QR = 11

Area of the
triangle will be calculated with Heron formula:

A =
sqrt[p(p-QP)(p-QR)(p-PR)]

where p =
(QR+QP+PR)/2

To calculate the height QM, we'll have to find
out the equation of PR , using the standard form y = mx+n, where m is the slope of PR.
After that, we'll consider the constraint the 2 lines are perpendicular if and only if
the product of  their slopes is -1.

To find the equation of
PR, we'll consider the formula:

(xR-xP)/(x-xP) =
(yR-yP)/(y-yP)

(5-1)/(x-1) =
(15-3)/(y-3)

4/(x-1) =
12/(y-3)

We'll divide by 4 both
sides:

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

We'll
cross multiply:

3x-3 =
y-3

We'll add 3 both sides:

3x
= y

So the slope of PR is m1 =
3

The slope of QM is m2 =
-1/3.

The equation of QM is:

y
- yQ = (-1/3)(x-xQ)

y-4 =
(-1/3)(x-5)