Dilations
What is a dilation?
A transformation that enlarges or reduces a figure proportionally from a center point.
What happens to a figure when the scale factor is greater than 1?
The figure enlarges (gets bigger).
If point A(2, 3) is dilated by scale factor 2, centered at origin, what is A′?
(4, 6)
What does it mean for two figures to be similar?
Same shape, different size; corresponding angles congruent, sides proportional.
Define slope.
The ratio of vertical change (rise) to horizontal change (run).
What is the center of dilation?
The fixed point about which a figure is dilated.
What happens to a figure when the scale factor is less than 1?
The figure reduces (gets smaller).
If point B is located at (–4, 2) and is dilated by ½, centered at origin, what is B′?
(–2, 1)
How do we know two triangles are similar?
Their corresponding sides are proportional and angles congruent.
The rise is 4 and run is 2. What is the slope?
Slope= 4/2 = 2.
True or False: Dilations change the angle measures of a figure.
False; dilations preserve angle measures.
If a triangle’s sides double in length, what is the scale factor?
Scale factor = 2.
If point C is located at(–2, –3) then is dilated by a factor of 3, what are the new coordinates?
(–6, –9)
How do similar triangles help us understand slope?
Slope uses ratios of similar right triangles.
A line rises 6 units while running 3 units to the right. Find its slope.
Slope= 6/3 = 2.
What stays the same during a dilation?
Angle measures and shape (only side lengths change).
A figure shrinks to half its size. What is the scale factor?
Scale factor = ½.
If point D was located at (1, 5) and D′ was located at (3, 15) after a dilation, what is the scale factor?
Multiply by 3 or Scale factor = 3.
Triangle ABC is dilated to Triangle DEF with scale factor ⅓. Which is larger?
Triangle ABC (original) is larger; scale factor < 1 shrinks the figure.
What is the slope of the line that goes down 3 and right 6?
Slope= -3/6 = -1/2
How do you perform a dilation centered at the origin?
Multiply each coordinate by the scale factor (x, y) → (k*x, k*y).
Explain how a scale factor of 0.25 affects a figure’s size.
The figure becomes ¼ of the original size (significant reduction).
If point E(–3, 4) is dilated and E′ is now located at point (–9, 12), what is the scale factor?
Multiply by 3 → Scale factor = 3.
What stays the same between two similar triangles?
Angles remain the same; side ratios are proportional.
How can slope be used to prove two lines are parallel?
Parallel lines have equal slopes.