Question

section 6: coordinate geometry

1. what are the coordinates of the origin?
2. find the distance between (2, 3) and (5, 7).
3. what is the midpoint between (0, 0) and (4, 4)?
4. what is the slope of a line passing through points (2, 4) and (6, 8)?
5. true or false: parallel lines have the same slope.

Explanation:

Step1: Recall origin definition

The origin is where \(x = 0\) and \(y = 0\).

Step2: Apply distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For \((x_1,y_1)=(2,3)\) and \((x_2,y_2)=(5,7)\), \(d=\sqrt{(5 - 2)^2+(7 - 3)^2}=\sqrt{9 + 16}=\sqrt{25}\)

Step3: Use midpoint formula \((\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\)

For \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(4,4)\), midpoint is \((\frac{0 + 4}{2},\frac{0+4}{2})=(2,2)\)

Step4: Use slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\)

For \((x_1,y_1)=(2,4)\) and \((x_2,y_2)=(6,8)\), \(m=\frac{8 - 4}{6 - 2}=\frac{4}{4}=1\)

Step5: Recall parallel lines property

By definition, parallel lines have the same slope.

Answer:

1. \((0,0)\)
2. \(5\)
3. \((2,2)\)
4. \(1\)
5. True