Lesson Guide
Explanation
The gradient of an interval measures how steep it is. Draw the right-angled triangle underneath it. The rise is how far you go up: from y equals two to y equals eight is six. The run is how far you go across: from x equals one to x equals five is four. The gradient is the rise divided by the run, six over four, which simplifies to three over two. As a formula, the gradient is y two minus y one, divided by x two minus x one. The interval goes uphill as you move to the right, so its gradient is positive.
Your Turn
Find the gradient from P(0, 1) to Q(3, 7). Check: rise 6, run 3, so m = 2.
Watch Out
Rise goes on top. Run over rise gives 4 ÷ 6 = \(\dfrac{2}{3}\), which is wrong.
Explanation
When an interval goes downhill from left to right, the rise is negative, so the gradient is negative. From C at negative two, five to D at four, one, the rise is one minus five, which is negative four, and the run is four minus negative two, which is six. The gradient is negative four over six, or negative two thirds. A horizontal line has a rise of zero, so its gradient is zero. A vertical line has a run of zero, and you cannot divide by zero, so its gradient is not defined.
Your Turn
Find the gradient from (1, 6) to (3, 0). Check: rise −6, run 2, so m = −3.
Watch Out
Subtract in the same order, top and bottom. (1 − 5) ÷ (−2 − 4) mixes the order and gives +\(\dfrac{2}{3}\), the wrong sign.
Explanation
The midpoint is exactly halfway between the two ends, so its coordinates are the averages. For E at negative three, four and F at five, negative two, add the x values: negative three plus five is two, and half of two is one. Add the y values: four plus negative two is two, and half of that is one. The midpoint is one, one. Check it on the grid: from E to M you go four across and three down, and from M to F you go the same four across and three down.
Your Turn
Find the midpoint of (2, 7) and (6, −1). Check: (4, 3).
Watch Out
Halve the sums. Adding without halving gives (2, 2), which is not even on the interval.
Explanation
The interval is the longest side of a right-angled triangle whose two short sides are the run and the rise. From G at one, one to H at seven, nine, the run is six and the rise is eight. By Pythagoras, G H squared equals six squared plus eight squared, which is thirty six plus sixty four, or one hundred. Take the square root: G H is ten. When the total is not a perfect square, leave the answer as a square root, such as root thirteen.
Your Turn
Find the length from (1, 2) to (3, 5). Check: 22 + 32 = 13, so the length is √13.
Watch Out
Square, add, THEN square-root. 6 + 8 = 14 is not the length, and neither is 100.
Explanation
Sometimes you know one end and the midpoint, and you need the other end. The midpoint is halfway, so the step from A to M happens again from M to B. From A at two, negative one to M at five, three, x goes up by three and y goes up by four. Take the same step from M: five plus three is eight, and three plus four is seven. So B is eight, seven. You can also use the midpoint rule: two plus x, halved, equals five, so x is eight. Quiz and exercises now.
Your Turn
A is (−1, 4) and M(1, 1) is the midpoint of AB. Find B. Check: the step is (+2, −3), so B is (3, −2).
Watch Out
Do not average A and M. That gives (3.5, 1), a point between them, not the far end.
📌 Key Points — this lesson in a nutshell
- Gradient m = rise ÷ run = \(\dfrac{y_2 - y_1}{x_2 - x_1}\). Uphill from left to right is positive; downhill is negative.
- A horizontal line has gradient 0. A vertical line has a gradient that is not defined (the run is 0).
- Midpoint = (average of the x values, average of the y values): add, then halve.
- Length: the run and the rise are the short sides of a right-angled triangle, so length2 = run2 + rise2.
- To find an end from the midpoint, take the step from the known end to M, then take it again.
Step 1: Introduction
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