The answers are;
- The distance between \((3,8)\) and \((7,-11)\) is \(19.42\text{ (to the nearest hundredth)}\)
- The midpoint of the line \(JK\) is \((6,-1)\)
- The midpoint of the line \(AB\) is \((8.5,9)\)
- The value of \(y\) is \(7\)
- The slope of the line that passes through \((3,-6)\text{ and }(6,12)\) is \(6\)
- The value of \(x\) is \(-1\)
- The value of \(y\) is \(-\frac{3}{2}\)
- The slope of the line that passes through \((8,7)\text{ and }(11,7)\) is \(0\)
In coordinate geometry, given two points \((x_1,y_1)\) and \((x_2,y_2)\);
- The distance between them is given by \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
- The midpoint has the coordinates \((\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\)
- The slope of the line passing through the points is the ratio \(\frac{y_2-y_1}{x_2-x_1}\)
These formulae will be used to solve the following problems:
1) Given points \((3,8)\) and \((7,-11)\), the distance between them is
\(\sqrt{(7-3)^2+(-11-8)^2}\approx 19.42 \text{ (to the nearest hundredth)}\)
2) If \(J=(-9,5) \text{ and }K=(21,-7)\), the midpoint of the line \(JK\) is
\((\frac{-9+21}{2}, \frac{5+(-7)}{2})=(6,-1)\)
3) If \(A=(-8,7) \text{ and } B=(-9,11)\), the midpoint of the line \(AB\) is
\((\frac{-8+(-9)}{2}, \frac{7+11}{2})=(8.5,9)\)
4) If the midpoint between \((8,y) \text{ and } (-11,6)\) is \((-1.5,5)\), then, using the
midpoint formula for the y-coordinate
\(6=\frac{y+5}{2} \implies y=7\)
5) The slope of the line that passes through \((3,-6)\text{ and }(6,12)\) is
\(\frac{12-(-6)}{6-3}=6\)
6) If the slope of the line that passes through \((x,10)\text{ and }(-4,8)\) is \(\frac{2}{3}\), then,
using the slope formula
\(\frac{2}{3}=\frac{8-10}{-4-x}\\\implies \frac{2}{3}=\frac{10-8}{4+x}\\\implies \frac{2}{3}=\frac{2}{4+x}\\\implies x=-1\)
7) If the slope of the line that passes through \((9,0)\text{ and }(3,y)\) is \(\frac{1}{2}\), then,
using the slope formula
\(\frac{1}{2}=\frac{y-0}{3-9}\\\implies \frac{1}{2}=\frac{y}{-3}\\\implies y=-\frac{3}{2}\)
8) The slope of the line that passes through \((8,7)\text{ and }(11,7)\) is
\(\frac{7-7}{11-8}=0\), (since the y-coordinates of both points are equal)
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