Question

Explaining Dilation and Scale Factor Facts
What do you know about the scale factor for the dilation shown? Explain.

Triangle A B C is reduced to triangle A prime B prime C prime.

Answer 1

I don't see the picture or numbers, so I can't give a single numeric scale factor. Here are the facts you can use to find and describe the scale factor for the dilation that reduces ΔABC to ΔA′B′C′:

1. Definition — scale factor k = (length of image)/(length of original).
k = A′B′/AB = B′C′/BC = C′A′/CA.

2. Because the triangle is reduced, 0 < k < 1. (All image side lengths are proportionally smaller.)

3. If the center of dilation is O, distances from O scale the same way:
k = OA′/OA = OB′/OB = OC′/OC.
If a vertex is the center (for example O = A), that vertex does not move (A′ = A) and the other distances still scale by k.

4. Orientation and parallelism — corresponding sides remain parallel and the triangles are similar (same angles). If k > 0 the orientation is preserved; if k were negative the image would be reflected through the center as well as scaled (but typical school reductions use positive k).

5. Coordinates — if the center is the origin and A(x,y) maps to A′(x′,y′), then x′ = kx and y′ = ky. For a center O = (ox,oy), use A′ = O + k(A − O).

6. Areas — area(ΔA′B′C′) = k^2 · area(ΔABC).

How to get the numeric k from your figure:
- Measure one pair of corresponding sides and compute k = (image side)/(original side), or
- Measure distances from the center of dilation: k = OA′/OA.

If you upload the diagram or give one pair of corresponding lengths or coordinates, I’ll compute the exact scale factor and show the work.