12.1 Homework: distance

Draw a point anywhere. Instead of labeling the specific coordinates of that point, just label it ( $x_1$ , $y_1$ ).

Draw another point somewhere else. Label it ( $x_2$ , $y_2$ ). To make life simple, make this point higher and to the right of the first point.

Draw the line going from ( $x_1$ , $y_1$ ) to ( $x_2$ , $y_2$ ). Then fill in the other two sides of the triangle

How far up is it from the first point to the second? (As always, start by thinking about specific numbers—then see if you can generalize.)

How far across is it from the first point to the second?

Find the distance $d$ from ( $x_1,y_1$ ) to ( $x_2,y_2$ ), using the Pythagorean Theorem. This will give you a general formula for the distance between any two points.

Plug in $x_2=0$ and $y_2=0$ into your formula. You should get the same formula you got on the previous assignment, for the distance between any point and the origin. Do you?

Draw a line from (0,0) to (4,10). Draw the point at the exact middle of that line. (Use a ruler if you have to.) What are the coordinates of that point?

Draw a line from (–3,2) to (5,–4). What are the coordinates of the midpoint?

Look back at your diagram of a line going from ( $x_1,y_1$ ) to ( $x_2,y_2$ ). What are the coordinates of the midpoint of that line?

Find the distance from the point (3,7) to the line $x=2$ .

Find the distance from the generic point ( $x$ , $y$ ) to the line $x=2$ .

Find the distance from the point (3,7) to the line $x=\mathrm{–2}$ .

Find the distance from the generic point ( $x$ , $y$ ) to the line $x=\mathrm{–2}$ .

Find the coordinates of all the points that have $y$ -coordinate 5, and which are exactly 10 units away from the origin.

Draw all the points you can find which are exactly 3 units away from the point (4,5).