Middle school maths, year 2 · unit 8
Similarity and Pythagoras
As the sun sinks, your shadow and the tree's shadow lengthen together. Know only that the two shadows change in the same proportion, and you can measure the tree without climbing it. Same shape, different size: that is similarity. We start by enlarging a triangle.
Bigger or smaller, the shape stays
A board to adjustEnlarge a photo and its size changes but its shape does not. Drag the gold corner of the teal triangle to enlarge and reduce it.
Just match two angles
A board to adjustWhat do you have to match to make a similar triangle? Turn the two handles of the stick triangle to set its angles.
Cut parallel, cut in the same ratio
A board to adjustCut a triangle parallel to its base: how are the two sides divided? Drag the gold point D.
Measuring height by shadow
A board to adjustYou can measure a tree without climbing it. Drag the sun to move the shadows.
Twice the length, four times the area
A board to adjustDouble the sides of a triangle: is the area doubled too? Drag the gold corner and count how many original pieces fit.
Three squares
A board to adjustA square is drawn on each side of a right-angled triangle. Change the two short sides with the gold handles and compare the three areas.
Making a right angle with 3, 4, 5
A board to adjustTwo sticks are pinned together at one end. Turn the teal stick until the third side is the right length. What angle do you get?
Try it yourself
Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.
What to take away from this unit
- If one figure can be enlarged or reduced by a fixed scale to fit exactly on another, the two are similar. All pairs of corresponding sides are in the same ratio (the scale ratio) and corresponding angles are equal.
- Two triangles are similar if two pairs of corresponding angles are equal (AA). They are also similar when three pairs of sides are in the same ratio (SSS), or two pairs of sides are in the same ratio with equal angles between them (SAS).
- A line parallel to one side of a triangle cuts the other two sides in the same ratio. The line joining the midpoints of two sides is parallel to the base and half its length.
- With scale ratio m : n, areas are in the ratio m² : n² and volumes in the ratio m³ : n³.
- The Pythagorean theorem: in a right-angled triangle the squares of the two short sides add up to the square of the hypotenuse. a² + b² = c².
- The other way round, if three sides satisfy a² + b² = c², the triangle is right-angled. 3, 4, 5 and 5, 12, 13 are such numbers.