0580 · 3.4Length and midpoint

Distance between two coordinates

0:00Imagine two locations marked on a coordinate grid. Their coordinates tell us where they are, but we can also calculate their distance apart. We can find the exact point halfway between them, too. Both calculations come from understanding horizontal and vertical movement, rather than memorising unrelated rules. A line segment is a straight part of a line joining two endpoints. Its length measures the distance between those endpoints. Its midpoint divides the segment into two equal lengths.

0:26We will calculate both using the coordinates of the endpoints. Remember that the x coordinate gives horizontal position, while the y coordinate gives vertical position. Subtracting coordinates tells us how much a position changes. We also need Pythagoras, which connects the three side lengths of a right-angled triangle. The hypotenuse squared equals the sum of the squares of the other two sides. Take point A at negative two comma one, and point B at four comma nine.

0:55Draw a horizontal line from A, then a vertical line to B. These form a right-angled triangle, with our original segment as its hypotenuse. The horizontal change is 4 − −2, giving six. For the vertical change, 9 − 1 gives eight. Pythagoras tells us that the segment length squared equals 6² + 8². Evaluating the squares gives 36 + 64, which equals one hundred. Taking √100 gives a length of ten units.

1:25Adding six and eight would instead give the distance along the two triangle sides. That route is longer than the straight segment. Pythagoras gives the direct distance because the horizontal and vertical movements meet at a right angle. The general rule follows exactly the same reasoning. Subtract the x coordinates and square that difference. Then square the difference between the y coordinates. Add those squares and take the square root to obtain the length.

1:50A coordinate change can be negative because it records direction. Length, however, cannot be negative. Squaring each change removes its sign before we add the results. Reversing the endpoints therefore produces the same distance, as it should.

Finding midpoints by averaging

2:04To reach the midpoint, travel halfway along the straight segment. This means completing half the horizontal change and half the vertical change. In our example, half of six is three, while half of eight is four. Starting at A, those movements take us to x equal to one and y equal to five. Another way to express halfway is to average the matching coordinates. Add negative two and four to obtain two.

2:28Dividing by two gives the midpoint x coordinate of one. For its y coordinate, 1 + 9 gives ten, and dividing by two gives five. Averaging works because the result lies equally far between the two starting values. We average the x coordinates separately from the y coordinates. Halving just one endpoint would find halfway between that point and the origin. It would not generally find halfway between our two endpoints.

2:53For a second example, take P at negative five comma four and Q at three comma four. Their vertical change is 4 − 4, giving zero. The segment is horizontal, so its length is 3 − −5, which equals eight units. There is no need to construct a triangle here. Adding the x coordinates gives negative two. Dividing that total by two gives negative one. For the y coordinate, 4 + 4 gives eight, and halving gives four.

3:20The midpoint is therefore the point negative one comma four. These methods must work without a calculator on Paper two. On Paper four, a calculator can evaluate square roots that are not whole numbers. Unless instructed otherwise, give answers that are not exact to three significant figures. Keep the full value during working and round only the final answer.

Exam-style distance and midpoint

3:41An exam-style question gives R at one comma negative two and S at six comma five. The horizontal change is 6 − 1, giving five. Subtracting negative two from five gives a vertical change of seven. The length is therefore the square root of five squared plus seven squared. The squared values give 25 + 49, totalling seventy-four. Its square root gives eight point six zero units to three significant figures.

4:06For the midpoint x coordinate, 1 + 6 gives seven, and dividing by two gives three point five. Adding negative two and five gives three, then halving gives a y coordinate of one point five. Common mistakes include losing a negative sign, mixing x and y coordinates, and forgetting the final square root. Subtracting a negative number means adding its positive value. Keep matching coordinates together, and remember that the squared distance is not the distance itself.

4:34For midpoint calculations, divide both coordinate totals by two. Length comes from a right-angled triangle formed by horizontal and vertical changes. Midpoint comes from taking half of each change, which is equivalent to averaging matching coordinates. A length is one distance, while a midpoint is a pair of coordinates. Those two ideas explain the methods and help you check your answers.