A triangle with vertices at (x1, y1) (x2, y2), and (x3, y3), as shown in the figure, has area eq

Name: Problem 46, Chapter 9: Precalculus
Uploaded: 2021-03-17T21:12:45-08:00
Duration: 2 min 36 s
Channel: James Kiss
Description: A triangle with vertices at (x1, y1) (x2, y2), and (x3, y3), as shown in the figure, has area equal to the absolute value of D, where D=(1)/(2)| x1 y1 1 x2 y2 1 x3 y3 1 | Find the area of each triangle having vertices at P, Q, and R. P(2,-2), Q(0,0), R(-3,-4)

step 1 I want to find the area of a triangle whose coordinates are at 2, negative 2, 0, 0, and negative

Question

A triangle with vertices at $\left(x_{1}, y_{1}\right)$ $\left(x_{2}, y_{2}\right),$ and $\left(x_{3}, y_{3}\right),$ as shown in the figure, has area equal to the absolute value of $D,$ where $$D=\frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\x_{2} & y_{2} & 1 \\x_{3} & y_{3} & 1\end{array}\right|$$ Find the area of each triangle having vertices at $P, Q,$ and $R$. $$P(2,-2), Q(0,0), R(-3,-4)$$

   A triangle with vertices at $\left(x_{1}, y_{1}\right)$ $\left(x_{2}, y_{2}\right),$ and $\left(x_{3}, y_{3}\right),$ as shown in the figure, has area equal to the absolute value of $D,$ where
$$D=\frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\x_{2} & y_{2} & 1 \\x_{3} & y_{3} & 1\end{array}\right|$$
Find the area of each triangle having vertices at $P, Q,$ and $R$.
$$P(2,-2), Q(0,0), R(-3,-4)$$

Precalculus

Margaret L. Lial,… 4th Edition

Chapter 9, Problem 46

Instant Answer

Solved by Expert James Kiss

03/17/2021

Step 1

We substitute these values into the determinant formula for the area of a triangle. The determinant $D$ is given by: $$D=\frac{1}{2}\left|\begin{array}{lll}2 & -2 & 1 \\0 & 0 & 1 \\-3 & -4 & 1\end{array}\right|$$ Show more…

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A triangle with vertices at $\left(x_{1}, y_{1}\right)$ $\left(x_{2}, y_{2}\right),$ and $\left(x_{3}, y_{3}\right),$ as shown in the figure, has area equal to the absolute value of $D,$ where $$D=\frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\x_{2} & y_{2} & 1 \\x_{3} & y_{3} & 1\end{array}\right|$$ Find the area of each triangle having vertices at $P, Q,$ and $R$. $$P(2,-2), Q(0,0), R(-3,-4)$$

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Key Concepts

Determinant Formula for Triangle Area

The area of a triangle with vertices provided in coordinate form can be efficiently calculated using the determinant of a 3×3 matrix. This formula is derived by expressing the coordinates of the triangle’s vertices in a matrix and evaluating its determinant, which yields twice the signed area of the triangle. Dividing by 2 provides the correct magnitude for the area, making the process straightforward when working with coordinate geometry.

Absolute Value and Signed Area

When using the determinant method, the result obtained is a signed area, meaning it can be negative depending on the order of the vertices. Since area is a non-negative quantity, taking the absolute value of the expression ensures that the final answer represents the true geometric area of the triangle. This step is crucial in formula applications, transforming the signed area into a proper measure of size.

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