Similarity & Proportionality — Challenge
Equal angles force proportional sides. That single fact measures buildings, mountains and stars.
What these problems 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.
Problem 1
Two triangles have two pairs of equal angles. What follows?
- They are similar
- They are congruent
- Nothing follows
- Their areas are equal
- They are right-angled
Worked answer
They are similar
If two pairs of angles are equal then the third pair must be too, because the angles of each triangle add to 180 degrees.
Equal angles throughout means the same shape, which is similarity.
It does not mean congruence, since the triangles may be different sizes. This is the AA test for similarity.
Problem 2
In two similar triangles, what is true of corresponding sides?
- They are equal
- They are proportional
- They are perpendicular
- They are parallel
- They are unrelated
Worked answer
They are proportional
Similar triangles have the same shape at possibly different sizes.
That means each side of one is the same multiple of the matching side of the other, so the sides are proportional.
Equal sides would make the triangles congruent, which is a special case.
Problem 3
One triangle has sides 2, 3, 4 and another has sides 4, 6, 8. How are they related?
- Congruent
- Similar, with ratio 1:2
- Not similar
- Similar, with ratio 1:4
- Similar, with ratio 2:1
Worked answer
Similar, with ratio 1:2
Compare matching sides: 24, 36 and 48 all equal 12.
Equal ratios throughout means the triangles are similar, with the first half the size of the second.
So the ratio from the first to the second is 1:2.
Problem 4
Two similar triangles have areas in the ratio 9:16. What is the ratio of their corresponding sides?
- 9:16
- 3:4
- 81:256
- 4:3
- 3:16
Worked answer
3:4
Areas of similar figures are in the ratio of the squares of the sides.
So the side ratio is the ratio of the square roots: √9:√16 = 3:4.
Squaring instead of taking roots would give 81:256.
Problem 5
A line parallel to one side of a triangle crosses the other two sides. How does it divide them?
- Proportionally
- Into equal parts
- Perpendicularly
- At right angles
- It does not divide them
Worked answer
Proportionally
The small triangle cut off shares an angle with the original, and the parallel line makes the other two angles equal as corresponding angles.
So the two triangles are similar by AA.
Similar triangles have proportional sides, so the two crossed sides are divided in the same ratio. Equal parts happens only if the line is a midline.
Problem 6
What do similar triangles always have equal?
- Their angles
- Their sides
- Their areas
- Their perimeters
- Nothing
Worked answer
Their angles
Similar means the same shape, possibly at a different size.
Shape is captured by the angles, so corresponding angles are equal.
Sides, areas and perimeters are all scaled by the size factor and so are generally unequal.
Problem 7
A triangle with a side of 6 is similar to one whose matching side is 9. What is the scale factor from the first to the second?
- 2
- 1.5
- 3
- 0.67
- 15
Worked answer
1.5
The scale factor is the new length divided by the old: 96.
96 = 1.5.
Dividing the other way gives about 0.67, which is the factor from the second triangle back to the first.
Problem 8
Two similar triangles have perimeters 12 and 18. What is the ratio of their corresponding sides?
- 3:2
- 2:3
- 4:9
- 9:4
- 1:6
Worked answer
2:3
A perimeter is a sum of lengths, so it scales by the same factor as the sides themselves.
The ratio is therefore 12:18, which simplifies to 2:3.
Areas would be in the ratio 4:9, the squares of these numbers.