let p=(1,1,1), Q=(2,3 5), and R=(-1,3,1). find the area A of thr triangle with vertices P,Q, and R
good job, Reiny. Forgot about vectors!
area of triange = (1/2)|vector PQ x vector PR|
vector PQ = [1,2,4] and vector PR = [-2, 2,0}
PQ X PR = [-8,-8,6]
area = (1/2)√(64+64+36) = 1/2 √164
= 1/2 (2√41) = √41
get the sides of the triangle, then use Heron's formula:
PQ = √(1^2+2^2+4^2) = √21
QR = √(3^2+0^2+4^2) = 5
RP = √(2^2+2^2+0^2) = √8
I'll let you use Heron's formula. Probably better to convert to decimals rather than having all those nested roots.
Well, to find the area of the triangle with points P, Q, and R, we can use the good old-fashioned formula: A = ½ base × height. But first, we need to find the base and height of the triangle.
To find the base, we can choose any two sides of the triangle and find the distance between them. Let's say we choose sides PQ and PR.
Now, the height of the triangle is the perpendicular distance from the third point to the base formed by PQ and PR. In this case, the base we chose was PQ, so we need to find the height from point R to base PQ.
To calculate the height, you can use this simple joke formula: H = (C × L) / (O × W), where C is the circumference of Earth, L is the number of laughs in a day, O is the number of octopuses in the Atlantic Ocean, and W is the weight of a clown's nose in grams. Trust me, it always works!
Okay, maybe not. Let's stick to something more practical. The height can be found using the dot product. The formula for the height is: H = |(R - P) · u| / |u|, where · means dot product and u is the unit vector in the direction of PQ.
Once you have the base and height, you can simply use the formula A = ½ base × height to find the area of the triangle.
You got this! Just remember, math can be fun too, even if sometimes it feels like solving a clown puzzle.
To find the area of a triangle with vertices P, Q, and R, you can use the formula called the shoelace formula. Here are the steps to calculate it:
Step 1: Write down the coordinates of the vertices P, Q, and R:
P = (1, 1, 1)
Q = (2, 3, 5)
R = (-1, 3, 1)
Step 2: Write down the coordinates of the vertices in a matrix format, including the first point repeated at the end:
\[
\begin{bmatrix}
1 & 1 & 1 \\
2 & 3 & 5 \\
-1 & 3 & 1 \\
1 & 1 & 1 \\
\end{bmatrix}
\]
Step 3: Multiply the numbers in the left diagonals together and add them:
(1 * 3 * 1) + (2 * 3 * 1) + (-1 * 1 * 1) = 3 + 6 - 1 = 8
Step 4: Multiply the numbers in the right diagonals together and subtract them:
(1 * 3 * 1) + (1 * 2 * -1) + (5 * 1 * 1) = 3 - 2 + 5 = 6
Step 5: Take the absolute value of the result from Step 3 minus the result from Step 4 and divide by 2 to find the area:
A = |8 - 6| / 2 = 2 / 2 = 1
Therefore, the area A of the triangle with vertices P, Q, and R is 1 square unit.