Question

lesson 5 - 1

1. calculate the distance between the points a(-4,2) and b(15,6).
2. calculate the distance between the points r(1.5,7) and s(-2.3,-8).
3. describe how to find the distance between two points on the coordinate plane.

Explanation:

Step1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Solve for the distance between \(A(-4,2)\) and \(B(15,6)\)

Here \(x_1=-4,y_1 = 2,x_2=15,y_2=6\).

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Step3: Solve for the distance between \(R(1.5,7)\) and \(S(-2.3,-8)\)

Here \(x_1 = 1.5,y_1=7,x_2=-2.3,y_2=-8\).

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Step4: Describe the distance - finding process

For two points \((x_1,y_1)\) and \((x_2,y_2)\) on the coordinate plane:

1. Subtract the \(x\) - coordinates: \(x_2-x_1\).
2. Subtract the \(y\) - coordinates: \(y_2 - y_1\).
3. Square each of the differences: \((x_2 - x_1)^2\) and \((y_2 - y_1)^2\).
4. Add the two squared results: \((x_2 - x_1)^2+(y_2 - y_1)^2\).
5. Take the square root of the sum to get the distance \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Answer:

1. The distance between \(A(-4,2)\) and \(B(15,6)\) is \(\sqrt{377}\approx19.42\).
2. The distance between \(R(1.5,7)\) and \(S(-2.3,-8)\) is \(\sqrt{239.44}\approx15.47\).
3. For two points \((x_1,y_1)\) and \((x_2,y_2)\) on the coordinate plane, use the formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), which involves subtracting \(x\) and \(y\) coordinates, squaring the differences, adding them, and then taking the square root.