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.

How do I actually proof the formula?

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

Ah, the lengths of a triangle, denoted by p, q, and r, have brought us here! Now, let me break down those formulas for you in my own clownish way:

First, we have the formula 1/2 pq sin R. This formula calculates the area of the triangle using the lengths p, q, and the angle measure R. The 1/2 is just a constant factor, the pq refers to the product of the two sides p and q, and sin R is the sine of the angle R.

Similarly, we have the formula 1/2 qr sin P. This formula calculates the area of the triangle using the lengths q, r, and the angle measure P. Again, the 1/2 is just playing around, qr refers to the product of the sides q and r, and sin P is the sine of the angle P.

Now, to justify the equation 1/2 pq sinR = 1/2 qr sin P, it's all about the area of the triangle. Since both formulas are calculating the same thing, which is the area, they must be equal! So, 1/2 pq sinR is equal to 1/2 qr sin P.

But wait! We can simplify this equation to make it more familiar. Canceling out the common factor of 1/2, we're left with pq sin R = qr sin P. And hey, look! That's pretty similar to another well-known formula: the Law of Sines! It states that a/sin A = b/sin B = c/sin C in any triangle.

So, my clownish friend, the simplified form of our equation is simply pq sin R = qr sin P, which looks just like the Law of Sines! Triangle trigonometry has a way of surprising us at times, doesn't it?

The formulas 1/2 pq sinR and 1/2 qr sin P are well known formulas in geometry used to calculate the area of a triangle. Let's break down their meaning and then justify the equation 1/2 pq sinR = 1/2 qr sin P.

1. 1/2 pq sinR: This formula represents the area of triangle PQR. It is calculated by taking half the product of the lengths of two sides PQ and QR and the sine of the included angle R. This formula is derived from the general formula for the area of a triangle, which is given by 1/2 base multiplied by height.

In this case, we consider PQ as the base and QR as the corresponding height. The included angle R determines the direction and magnitude of the height. Multiplying 1/2 by pq and sinR gives us the area of triangle PQR.

2. 1/2 qr sin P: Similarly, this formula represents the area of triangle PQR. It is calculated by taking half the product of the lengths of two sides QR and RP and the sine of the included angle P. Here, QR is considered as the base and RP as the corresponding height. The included angle P determines the direction and magnitude of the height.

Now, let's justify the equation 1/2 pq sinR = 1/2 qr sin P:

We start with the two area formulas as mentioned above: 1/2 pq sinR and 1/2 qr sin P. Since both formulas represent the area of the same triangle, they should give us the same result. Mathematically,

1/2 pq sinR = 1/2 qr sin P

Now, we can simplify this equation. First, canceling the common factor of 1/2, we get:

pq sinR = qr sin P

Finally, we can rearrange the equation to a more familiar form, which is known as the Law of Sines. The Law of Sines states that in any triangle with sides a, b, c, and opposite angles A, B, C, respectively, the following relationship holds:

a / sin A = b / sin B = c / sin C

In our equation, we have the sides and angles labeled differently, but we can match them to the Law of Sines form:

pq / sin R = qr / sin P

This equation demonstrates the Law of Sines, where pq, qr, and the corresponding angles R and P form a relationship.

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.