Geometrical problems on coordinate axes

Swing line B until it is square to line A

Red line A has gradient 2. It stays put. Line B starts parallel to it, because the two gradients are equal. Swing B round: when B’s gradient is −1/2 the product reads −1.00 and the lines cross at a right angle. At any other gradient the product is not −1 and the angle is not square.

Both axes use the same scale. Lift or lower the dot to swing line B round.

Maths · Dividing a segment

A is (2, 1) and B is (10, 13). Drag P along AB and watch the ratio, the x distance and the y distance move together.

Maths · Shapes on axes

What does this evidence prove?

Pick a piece of evidence about a quadrilateral, then choose the shape it proves. Some evidence proves nothing yet.

6 of 6 still to sort.

What you need to know

• Gradient is the change in y divided by the change in x, both worked out the same way round.
• Parallel lines have the same gradient.
• Have a goYour mate swears the line through (1, 2) and (3, 8) is parallel to a line with gradient 2, because they “look about the same”. Do the numbers agree?

  No. The gradient is (8 − 2) ÷ (3 − 1) = 3, not 2.

  Parallel means exactly the same gradient. “Looks about the same” is how a gradient of 3 sneaks past a gradient of 2.

• Perpendicular lines have gradients that multiply to −1: negative reciprocals, like 2 and −1/2.
• Have a goA line has gradient 5. What gradient does a line perpendicular to it have?

  −1/5

  Flip 5 to get 1/5, then change the sign. Flipping alone gives 1/5, and 5 × 1/5 is +1, which is not the −1 that perpendicular needs.

• For a perpendicular line through a point, use the negative reciprocal as m, then substitute the point into y = mx + c.
• A perpendicular bisector is perpendicular to the segment and passes through its midpoint: halve the x and y journeys from A.
• AP : PB = 1 : 3 gives 4 equal parts, so P is a quarter of the way from A.
• Have a goSam sees AP : PB = 1 : 4 and says, “P is a quarter of the way along AB.” Where does Sam go wrong?

  1 + 4 = 5 equal parts, so P is one fifth of the way from A.

  The ratio compares the two pieces, AP and PB. The whole of AB is both pieces together, so add them to count the equal parts.

• Lines only look perpendicular when the axes have a 1 : 1 scale, so check the scales before trusting a drawing.
• Have a goYou plot lines with gradients 2 and −1/2 on axes whose scales are not in the ratio 1 : 1. Will the drawn lines look perpendicular?

  No. On skewed scales the drawn lines will not look perpendicular.

  Their gradients multiply to −1, but gradients read from a drawing on skewed axes are not negative reciprocals, so the picture stops showing it. Check the scales first.

• Two pairs of equal opposite-side gradients make a parallelogram; add adjacent gradients multiplying to −1 and it is a rectangle.
• A parallelogram with perpendicular diagonals is a rhombus, but perpendicular diagonals alone are not enough: a kite has them too.
• To prove a shape’s type, show every step and explain it; to disprove it, one failed property is enough.

The big picture

On coordinate axes, geometry turns into arithmetic. Gradients decide whether lines are parallel or perpendicular, midpoints and ratios place points on a segment, and those same tools prove which quadrilateral a set of points makes.

⚠ Watch out

Trusting the picture. Lines that look parallel or square, or diagonals that look perpendicular, prove nothing. Calculate the gradients and midpoints, and show each property the shape needs.

Memory hook

Perpendicular: flip it and flip its sign. 2 becomes −1/2.

Check yourself

Close the page and say from memory how you test two lines for parallel, how you test for perpendicular, and what you check about the axes before you trust a diagram.