Coordinate Geometry: Lines, Lengths, and Midpoints

Worked Example: Deriving a Line's Equation

Worked Example: The Line's 'Playlist' 🎧

1. 1
   First, let's find the gradient ($m$). We'll use our formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Let A be $(x_1, y_1)$ and B be $(x_2, y_2)$.

   $m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3$
2. 2
   Now we know the line's equation looks like $y = 3x + c$. To find $c$, we can substitute the coordinates of either point into the equation. Let's use point A(2, 5), so $x=2$ and $y=5$.

   $5 = 3(2) + c$
3. 3
   Solve for $c$.

   $5 = 6 + c \implies c = 5 - 6 = -1$
4. 4
   We've got our gradient ($m=3$) and our y-intercept ($c=-1$). Now just write the final equation.

   $y = 3x - 1$

Worked Example: Calculating Segment Length

Worked Example: The Distance Drop 📍

1. 1
   Let's label our points: C is $(x_1, y_1)$ so $x_1 = -1, y_1 = 3$. D is $(x_2, y_2)$ so $x_2 = 5, y_2 = -2$. Now plug these into the distance formula.

   $d = \sqrt{(5 - (-1))^2 + (-2 - 3)^2}$
2. 2
   Be careful with the negative numbers! $5 - (-1)$ is $5+1=6$, and $-2-3 = -5$.

   $d = \sqrt{(6)^2 + (-5)^2}$
3. 3
   Now, square both numbers inside the square root. Remember, a negative number squared becomes positive.

   $d = \sqrt{36 + 25} = \sqrt{61}$
4. 4
   The question doesn't ask for a decimal, so leaving it as a square root (a 'surd') is often the most accurate answer. If you need to, you can calculate the decimal value.

   $d = \sqrt{61} \approx 7.81 \text{ (to 3 s.f.)}$

Worked Example: Finding the Midpoint

Worked Example: The Halfway Hangout Spot ☕

1. 1
   Let's identify our coordinates. For P, $x_1 = -2$ and $y_1 = -4$. For Q, $x_2 = 6$ and $y_2 = 10$.

   $M = \left( \frac{-2 + 6}{2}, \frac{-4 + 10}{2} \right)$
2. 2
   Now, just calculate the value for each coordinate separately.

   $M = \left( \frac{4}{2}, \frac{6}{2} \right)$
3. 3
   Simplify the fractions to get the final coordinates of the midpoint.

   $M = (2, 3)$

Worked Example: Equation of a Perpendicular Bisector

Worked Example: The Final Boss Battle ⚔️

1. 1
   Step 1: Find the midpoint of AB. We need this point because the bisector passes through it.

   $M = \left( \frac{1 + 5}{2}, \frac{2 + 10}{2} \right) = \left( \frac{6}{2}, \frac{12}{2} \right) = (3, 6)$
2. 2
   Step 2: Find the gradient of the line segment AB.

   $m_{AB} = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2$
3. 3
   Step 3: Find the perpendicular gradient. We flip the gradient of AB and switch the sign. The gradient of AB is 2 (or $\frac{2}{1}$).

   $m_{\perp} = -\frac{1}{2}$
4. 4
   Step 4: Now we have the gradient of our new line ($-\frac{1}{2}$) and a point it passes through (the midpoint, (3, 6)). We use $y = mx + c$ to find $c$.

   $6 = -\frac{1}{2}(3) + c \implies 6 = -1.5 + c \implies c = 7.5$
5. 5
   Finally, write the equation of the perpendicular bisector using our new gradient and new y-intercept.

   $y = -\frac{1}{2}x + 7.5$

Worked Example: Equation of a Perpendicular Bisector

Let's Solve It: The Perpendicular Bisector Boss Level 👾

1. 1
   First, we need to find the dead center of the line AB. This midpoint is the one point we know for sure is on our perpendicular bisector. We'll use the midpoint formula.

   $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-2 + 4}{2}, \frac{1 + 3}{2} \right) = \left( \frac{2}{2}, \frac{4}{2} \right) = (1, 2)$
2. 2
   Next, let's figure out the slope of the original line segment AB. This tells us the 'direction' we need to be perpendicular to.

   $m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}$
3. 3
   Our bisector is perpendicular to AB, so its gradient is the negative reciprocal of AB's gradient. Time to flip the fraction and switch the sign!

   $m_{\perp} = -\frac{1}{m_{AB}} = -\frac{1}{(1/3)} = -3$
4. 4
   We have our perpendicular gradient ($m = -3$) and a point on the line (the midpoint $M(1, 2)$). Let's plug these into $y = mx + c$ to find our y-intercept.

   $2 = (-3)(1) + c <br> 2 = -3 + c <br> c = 5$
5. 5
   We've got our gradient and our y-intercept. Mission complete! Let's write out the final equation for the perpendicular bisector.

   $y = -3x + 5$

Worked Example: Finding the Equation from Two Points

Let's Build This Equation from Scratch 🛠️

1. 1
   First, we need to find the gradient ($m$). Think of this as figuring out the difficulty setting. We'll use the gradient formula and label our points $A(x_1, y_1)$ and $B(x_2, y_2)$.

   $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 1}{4 - (-2)} = \frac{9}{6} = \frac{3}{2}$
2. 2
   Now we have the gradient, $m = \frac{3}{2}$. Our equation looks like $y = \frac{3}{2}x + c$. To find $c$, we can substitute the coordinates of either point A or B into the equation. Let's use point B(4, 10), so we'll sub in $x=4$ and $y=10$.

   $10 = \left(\frac{3}{2}\right)(4) + c$
3. 3
   Solve the equation for $c$. This is the final puzzle piece we need.

   $10 = 6 + c \implies c = 10 - 6 \implies c = 4$
4. 4
   We've got our gradient ($m$) and our y-intercept ($c$). Now we just assemble them into the final form $y = mx + c$. Mission complete!

   $y = \frac{3}{2}x + 4$

Worked Example: Calculating Gradient from Two Coordinates

From Game Levels to Gradients 🎮

1. 1
   First, let's identify our two points from the problem. We can treat time as our $x$-coordinate and the game level as our $y$-coordinate. So, our two points are $(2, 5)$ and $(6, 17)$. Let's label them to avoid any mix-ups.

   $(x_1, y_1) = (2, 5) \text{ and } (x_2, y_2) = (6, 17)$
2. 2
   Next, we'll write down the gradient formula. This is our key tool for solving the problem. Always a good idea to write it down first to get it locked in your brain.

   $m = \frac{y_2 - y_1}{x_2 - x_1}$
3. 3
   Now, we just substitute our values into the formula. Make sure the $y$-values are on top in the numerator and the corresponding $x$-values are on the bottom in the denominator.

   $m = \frac{17 - 5}{6 - 2}$
4. 4
   Time for the easy part – do the subtraction on the top and bottom to simplify the fraction.

   $m = \frac{12}{4}$
5. 5
   Finally, we calculate the final value. This number represents the gradient. So, what does it actually mean in this context?

   $m = 3$
   This means for every 1 hour you play, you gain 3 levels. That's a solid rate of progress!

Worked Example: Midpoint and Distance Calculation

The Quest for the Halfway Point 🤝

1. 1
   First, let's identify our coordinates so we don't mix them up. It doesn't matter which point is 1 or 2, as long as you're consistent. Let's set Town A as $(x_1, y_1)$ and Town B as $(x_2, y_2)$.

   Let $A = (x_1, y_1) = (-4, -2)$
   Let $B = (x_2, y_2) = (8, 3)$
2. 2
   For part (a), we need to find the midpoint. We'll use the midpoint formula by plugging in our x and y values. This gives us the exact coordinates of that snack stop!

   $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-4 + 8}{2}, \frac{-2 + 3}{2} \right) = \left( \frac{4}{2}, \frac{1}{2} \right) = (2, 0.5)$
3. 3
   So, the service station is at coordinates $(2, 0.5)$. Now for part (b), we need the total distance. We'll use the distance formula, which is just Pythagoras in disguise. Let's plug our coordinates into it.

   $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(8 - (-4))^2 + (3 - (-2))^2}$
4. 4
   Be careful with the double negatives! 'Minus a negative' becomes a positive. Now we simplify the brackets, square the results, add them, and finally take the square root.

   $d = \sqrt{(8 + 4)^2 + (3 + 2)^2} = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169}$
5. 5
   The final step is to calculate the square root to get our answer. This gives us the total distance of the trip.

   $d = 13$
   The total distance is 13 units.

Worked Example: Plotting $y = -\tfrac{1}{2}x + 4$ from a Table

1. 1
   Substitute each $x$ into the equation. Fill the table.

   $x = -2: y = 1 + 4 = 5$ $x = 0: y = 0 + 4 = 4$ $x = 2: y = -1 + 4 = 3$ $x = 4: y = -2 + 4 = 2$ $x = 6: y = -3 + 4 = 1$
2. 2
   Check the pattern: every time $x$ goes up by $2$, $y$ goes down by $1$. That's the gradient as rise over run.

   $m = \frac{\Delta y}{\Delta x} = \frac{-1}{2} = -\tfrac{1}{2}$
3. 3
   The $y$-intercept is just the $y$-value when $x = 0$ — grab it from the table.

   $y\text{-intercept} = 4, \quad \text{line passes through } (0, 4)$
4. 4
   Plot the five points, rule the line, write the summary. 🎯

   $\text{Gradient} = -\tfrac{1}{2}, \quad y\text{-intercept} = 4$