Similarity & Proportionality — Print

What the practice questions cover

These questions use 21 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

• Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
• Scale factor — The number every length is multiplied by.
• Similar figures — Same shape, different size: equal angles, proportional sides.
• Perimeter — Distance all the way around the outside.
• Area — Space inside, measured in square units.
• Volume — Space inside, in cubic units.
• Base — The number being multiplied. In 53 the base is 5.
• Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
• Scale — What one grid step is worth — and where the axis starts.
• Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
• Similar — Same shape, possibly different size. Written △ ABC ∼ △ DEF.
• Parallel — Same slope, different intercept.
• Ray — One endpoint and everything beyond it in one direction. Written AB.
• Prove — The statement you must arrive at. Always the last line.
• Included angle — The angle between two named sides. The A in SAS must be included.
• AA similarity — Two pairs of congruent angles prove two triangles similar.
• SSS similarity — All three pairs of sides in the same ratio.
• SAS similarity — Two pairs of sides in ratio with the included angles congruent.
• Geometric mean — For a, x, b in proportion, x = √ab.
• Pole — The origin, where r = 0.
• Height — The perpendicular distance from the base to the opposite corner.

Why two angles are enough

A triangle has three angles that add to 180°. Fix two of them and the third has no choice.

So two matching angles guarantee all three match, and the triangles are similar. That is AA, and it is the test you will use most.

Note the contrast with congruence: AAA is useless there, because it says nothing about size. For similarity, size is exactly what we are not fixing.

The side-based tests

SSS similarity: all three pairs of sides in the same ratio. Compute all three and check they agree.

SAS similarity: two pairs in ratio, with the angles between them congruent. The word "included" matters exactly as it did for congruence.

Use these when the question gives lengths rather than angles.

The triangle proportionality theorem

A line parallel to one side of a triangle divides the other two sides proportionally.

If the line cuts one side into a and b, and the other into c and d, then ab = cd.

Its converse is also true: if a line cuts two sides proportionally, it must be parallel to the third.

Indirect measurement

A person 5 ft tall casts a 4 ft shadow; a tree casts a 32 ft shadow at the same moment.

The sun's rays arrive parallel, so the two triangles have the same angles, so they are similar.

54 = h32 gives h = 40 ft. You measured a tree with a tape measure and no ladder.

Perimeter, area and volume

If the scale factor is k, then perimeters are in the ratio k, areas in the ratio k², and volumes in the ratio k³.

Two similar triangles with sides in ratio 2:3 have areas in ratio 4:9.

Going backwards, areas in ratio 9:16 means sides in ratio 3:4 — take the square root.