KS4. Algebra. Length, Midpoint & Gradient

Below are the formulae for finding:

the distance between two points (i.e. the length of the line segment that joins them);
The midpoint of two points; and
The gradient of the line segment between two points.

In each of the formulae, the two points are (x₁,y₁) and (x₂,y₂) (each of these letters with a subscript refers to a specific number which will be in the question).

Length: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Midpoint: $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Gradient: $\frac{y_2-y_1}{x_2-x_1}$ .

We can also find the equation of the line, using the formula y-y₁=m(x-x₁), where the letters x and y are variables, i.e. they stay as letters in our final equation, and all of the other letters are constants, i.e. we put a specific number into them for each of our questions.

If two lines are perpendicular, we can find the gradient of the perpendicular line using the formula m x n= -1, where m is the formula of one line and n is the formula of the perpendicular line.

Exercise

Now let’s use the equations with the following points to answer the following questions:

A: (2,4), B: (1,3), C: (-1,5), D: (8,-1), E: (-2,-4), F: (-9,3), G: (-8,-8)

Find the length, midpoint, gradient and equation of the following lines. All “square roots” answers can be left as surds (e.g. $\sqrt{3}$ ) and all equations should be put into the form y=mx+c (note that the exercise will be more beneficial in helping you learn the formula if you do the questions one at a time (i.e. not calculating the lengths for all of the questions first, etc.):

(1.) AB, (2.) AC, (3.) AD, (4.) AE, (5.) AF, (6.) AG, (7.) BC, (8.) BD, (9.) BE, (10.) BF, (11.) BG, (12.) CD, (13.) CE, (14.) CF, (15.) CG, (16.) DE, (17.) DF, (18.) DG, (19.) EF, (20.) EG

Answers

AB: Length: $\sqrt{2}$ , Midpoint: (1.5,3.5), Gradient: 1, Equation: y=x+2
AC: Length: $\sqrt{10}$ , Midpoint: (0.5,4.5), Gradient: -1/3, Equation: y=-⅓x+4⅔
AD: Length: $\sqrt{61}$ , Midpoint: (5,1.5), Gradient: -5/6, Equation: $y = -\frac{5}{6}x + 4\frac{5}{3}$
AE: Length: $\sqrt{80} = 4\sqrt{5}$ , Midpoint: (0,), Gradient: 2, Equation: y=2x
AF: Length: $\sqrt{122}$ , Midpoint: (-3.5,3.5), Gradient: 1/11, Equation: $y=\frac{1}{11}x + 3\frac{9}{11}$
AG: Length: $\sqrt{244} = 4\sqrt{61}$ , Midpoint: (-3,-2), Gradient: $1\frac{1}{5}$ , Equation: $y=1 \frac{1}{5}x + 1 \frac{3}{5}$
BC: Length: $\sqrt{8} = 2\sqrt{2}$ , Midpoint: (0,4), Gradient: -1, Equation: y=-x+4
BD: Length: $\sqrt{65}$ , Midpoint: (4.5,1), Gradient: -4/7, Equation: $y=-\frac{4}{7}x + 3\frac{4}{7}$
BE: Length: $\sqrt{58}$ , Midpoint: (-0.5,-0.5), Gradient: $2\frac{1}{3}$ , Equation: $y=\frac{7}{3}x + \frac{2}{3}$
BF: Length: 10, Midpoint: (-4,3), Gradient: 0, Equation: y=3
BG: Length: $\sqrt{202}$ , Midpoint: (-3.5,-2.5), Gradient: $1\frac{2}{9}$ , Equation: $y=1\frac{2}{9}x + 1\frac{7}{9}$
CD: Length: $\sqrt{117} = 3\sqrt{13}$ , Midpoint: (3.5,2), Gradient: -⅔, Equation: $y=-\frac{2}{3}x+4\frac{1}{3}$
CE: Length: $\sqrt{82}$ , Midpoint: (-1.5,0.5), Gradient: 9, Equation: y=9x+14
CF: Length: $\sqrt{68} = 2\sqrt{17}$ , Midpoint: (-5,4), Gradient: 1/4, Equation: $y=\frac{1}{4}x+5\frac{1}{4}$
CG: Length: $\sqrt{218}$ , Midpoint: (-4.5,1.5), Gradient: $1\frac{6}{7}$ , Equation: $y=\frac{13}{7}x + 6\frac{6}{7}$
DE: Length: $\sqrt{109}$ , Midpoint: (3,-2.5), Gradient: 3/10, Equation: $y=/frac{3}{10}x-3/frac{2}{5}$
DF: Length: $\sqrt{305}$ , Midpoint: (-0.5,1), Gradient: -4/17, Equation: $y=-\frac{4}{17}x+\frac{15}{17}$
DG: Length: $\sqrt{305}$ , Midpoint: (0, -4.5), Gradient: 7/16, Equation: $y=\frac{7}{16}x - 4\frac{1}{2}$
EF: Length: $\sqrt{98}=7\sqrt{2}$ , Midpoint: (-5.5,-0.5), Gradient: -1, Equation: y=-x-6
EG: Length: $\sqrt{52}=2\sqrt{13}$ , Midpoint: (-5,-6), Gradient: 2/3, Equation: $y= \frac{2}{3}x-2 \frac{2}{3}$

As an additional exercise, please find the perpendicular bisector for each of the line segments specified about (ask your teacher how to do this if you are not sure).

The answers are below:

1.) y = -x + 5

2.) y = 3x + 3

3.) y = 1.2x – 4.5

4.) y = -0.5x

5.) y = 11x + 42

6.) y = (-5/6)x – 4.5

KS4. Algebra. Length, Midpoint & Gradient

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