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The distance formula is essential for finding the length of a line segment between two points on a coordinate plane. Imagine you have two points, \((x_1, y_1)\) and \((x_2, y_2)\). The distance between these points is calculated as \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). This formula is derived from the Pythagorean theorem and helps in determining the straight-line distance between any two given points in a 2-dimensional space.
For example, in our triangle problem, to find the side \(AB\) between points \((-1,1)\) and \((3,1)\), we plug into the formula:

• \((x_2-x_1)\) becomes \((3 - (-1))\) which equals 4.
• \((y_2-y_1)\) becomes \((1-1)\) which equals 0.

This gives us \(\sqrt{4^2 + 0^2} = 4\). By repeating this process for other sides, you can easily find the lengths of \(BC\) and \(CA\).
Understanding how to use this formula efficiently is key in solving various geometrical problems.

Calculating the area of a triangle when you know the vertices' coordinates requires a special formula. This method is especially useful when the triangle is plotted on a coordinate grid. The area is calculated using: \[\text{Area} = \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|\]This formula works by effectively breaking down the coordinates into manageable parts that calculate the positive and negative parts of the enclosed area.
Following our triangle's vertices \((-1, 1)\), \((3, 1)\), and \((3, -2)\), we substitute each value:

• For \(x_1(y_2-y_3)\), compute \((-1)(1-(-2)) = -3\).
• For \(x_2(y_3-y_1)\), compute \((3)(-2-1) = -9\).
• For \(x_3(y_1-y_2)\), compute \((3)(1-1) = 0\).

Adding these values gives \(-3 - 9 + 0 = -12\). Taking the absolute value and halving it gives us the area \(= 6\).
With practice, this method becomes a straightforward way to tackle problems involving coordinate geometry.

The perimeter of a triangle is simply the sum of the lengths of all its sides. It represents the total distance around the triangle, which is helpful in various practical applications like fencing a triangular plot.
To find the perimeter of a triangle built from points on a coordinate plane, you first use the distance formula to find each side's length. With known sides:

• Side \(AB = 4\)
• Side \(BC = 3\)
• Side \(CA = 5\)

You then add these lengths: \[\text{Perimeter} = 4 + 3 + 5 = 12\]Simple, right? The formula works consistently for any triangle's coordinate points, making it a reliable method for problems in coordinate geometry. Knowing how to derive these side lengths quickly allows you to easily calculate perimeter and can also lead you into exploring deeper properties of geometric shapes.