Question

The lengths QR, RP, and PQ in triangle PQR are often denoted p, q, and r, respectively.
What do the formulas 1/2 pq sinR and 1/2 qr sin P mean? After you justify the
equation 1/2 pq sinR = 1/2 qr sin P, simplify it to a familiar form.

Answer 1

The formulas represent the area of the triangle

Did you want an actual proof of the formula?

Hint: draw a perpendicular from P to QR, call it h
take sinR, then find the area by (1/2)base*height.

Answer 2

How do I actually proof the formula?

Answer 3

ok, follow my steps above

you now have a right-angled triangle with a height of h
sin R = h/PR = h/q
h = qsin R

Isn't the area of the triangle (1/2)(base)h
= (1/2)QRh
= (1/2)p(qsin R)
= (1/2)pq sin R as requested.

dropping perpendiculars from R and Q you can prove in the same way that

area = (1/2)rq sinP and (1/2)rpsinQ