Topic A: Dilations
Lessons 1–5 · Scale factor, center of dilation, and properties.
01–03 — Lessons 1–3
Dilations & Scale Factor
- Dilation: a transformation that enlarges or shrinks a figure by a scale factor k, from a center point.
- Scale factor k > 1: enlargement. 0 < k < 1: reduction.
- The image is similar (same shape) to the pre-image, not congruent (unless k=1).
- Dilation from origin: multiply both coordinates by k. (x,y) → (kx, ky).
- Distances scale by k; areas scale by k².
Dilation with k=3 from origin: A(2,4) → A'(6,12). Side length triples; area × 9.
Practice — Lessons 1–3
Q1 · Multiple Choice
A dilation with scale factor 1/2 from the origin maps (8, 6) to:
Q2 · Multiple Choice
If a dilation has scale factor k = 3, how does the area change?
04–05 — Lessons 4–5
Properties of Dilations
- A dilation maps a line through the center of dilation to itself.
- A line NOT through the center maps to a parallel line.
- Angle measures are preserved under dilation.
- The ratio of any two corresponding lengths = k (the scale factor).
- Dilations produce similar figures: same shape, different (or equal) size.
△ABC dilated by k=2 to △A'B'C'. All angles equal; all sides doubled. Perimeter doubles; area ×4.
Practice — Lessons 4–5
Q3 · Multiple Choice
After a dilation, which is preserved?
Q4 · Multiple Choice
A line through the center of dilation maps to:
Topic B: Similarity Criteria
Lessons 6–11 · AA, SAS~, SSS~ and proving similarity.
06–08 — Lessons 6–8
Triangle Similarity Shortcuts
- AA (Angle-Angle): two pairs of congruent angles → triangles similar.
- SAS~ (Side-Angle-Side): two sides proportional and included angle ≅ → similar.
- SSS~ (Side-Side-Side): all three pairs of sides proportional → similar.
- Similar triangles: ∠s congruent, sides proportional (ratio = scale factor).
- Notation: △ABC ∼ △DEF (order matters — corresponding vertices listed).
△ABC ∼ △DEF with k = 2: if AB = 4, then DE = 8. If ∠A = 50°, then ∠D = 50°.
Practice — Lessons 6–8
Q5 · Multiple Choice
Two triangles each have angles 40° and 70°. By AA, they are:
Q6 · Multiple Choice
For SAS~ similarity, you need:
09–11 — Lessons 9–11
Applying Similar Triangles
- Set up proportions using corresponding sides of similar triangles.
- Cross-multiply to solve for unknown lengths.
- Altitude to hypotenuse: creates two smaller triangles, each similar to the original.
- Geometric mean: altitude h = √(p·q), where p, q are the two segments of the hypotenuse.
- Use similar triangles to find indirect measurements (shadow, mirror methods).
△ABC ∼ △DEF. AB/DE = BC/EF = AC/DF. If AB=6, DE=9, BC=8 → EF = 8·(9/6) = 12.
Practice — Lessons 9–11
Q7 · Short Answer
△ABC ∼ △DEF. AB = 4, DE = 10, BC = 6. Find EF.
Q8 · Multiple Choice
You use a mirror on the ground to find a tree's height. The similar triangles method relies on:
Topic C: Proportional Relationships
Lessons 12–18 · Triangle proportionality theorem and other results.
12–18 — Lessons 12–18
Proportionality Theorems
- Triangle Proportionality Theorem: a line parallel to one side of a triangle divides the other two sides proportionally.
- Converse: if a line divides two sides proportionally, it is parallel to the third.
- Midsegment: connects midpoints of two sides; parallel to the third side and half as long.
- Angle bisector theorem: the bisector of an angle divides the opposite side proportionally.
- Side-splitter: if two triangles share a common angle and their sides are proportional, they are similar.
In △ABC, midsegment MN (M on AB, N on AC): MN ∥ BC and MN = BC/2.
Practice — Lessons 12–18
Q9 · Multiple Choice
The midsegment of a triangle is:
Q10 · Short Answer
In a triangle, a midsegment is 7 cm. How long is the parallel base?