Use equal angles and proportional sides to distinguish similarity from congruence.
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Try
A triangle with base 4 and perpendicular height 2 has area 4. Dilating it by factor 2 about one vertex produces base 8, height 4 and area 16. The length factor is 2; the area factor is 4.
Predict
Predict what stays fixed under a positive dilation. Angles remain equal, and corresponding lengths share one scale factor k. A dilation needs a center O: the image P′ lies on ray OP with OP′=k·OP. A line not through O maps to a parallel line; a line through O remains on that same line.
Experiment
Use the nested triangles in the diagram to compare lengths and areas. Write the factor as image length divided by original length. Then change the center on paper: the resulting triangle has the same size as before but a different location. Doubling coordinates works only when the dilation center is the origin.
A dilation of factor 2 about A doubles both perpendicular legs. The larger triangle has four times the area. Corresponding lengths are drawn proportionally.
Understand
Worked argument: in triangle ABC, points D on AB and E on AC satisfy AD/AB=AE/AC=1/3. The angle at A is shared, so the small and large triangles are similar by proportional sides with included angle (SAS similarity). Therefore DE/BC=1/3. The correspondence also makes the matching base angles equal; the parallel-angle converse gives DE∥BC.
Invalid argument: one pair of proportional sides proves similarity. Any two positive lengths have some ratio, so that single condition imposes no shape constraint. Use AA, all three corresponding side ratios, or two side ratios with the included angle.
Apply
Match corresponding vertices before writing a ratio. Keep every ratio in the same direction. A figure can be similar without being congruent; only scale factor 1 preserves corresponding lengths. Your area argument should account for both base and perpendicular height.
Continue
Depth: write the coordinate rule P′=O+k(P−O) component by component. For k>0 the image stays on the same ray. Negative factors combine a positive dilation with a half-turn, and k=0 collapses the figure; this lesson’s ordinary similarity statements assume k>0. The scaling investigation compares linear, area and volume factors; the volume extension is separate from the planar proof here.
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Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Similar triangles have corresponding sides 6 and 15. A second side of the smaller triangle is 8. Find the matching larger side.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A dilation has factor 3/2. A region originally has area 20 cm². Find the new area in cm².
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Two nondegenerate triangles have two corresponding angles equal. Which conclusion follows?
Dilate point (5,3) by factor 2 about center (1,1). What is the image?
In triangle ABC, D and E are the midpoints of AB and AC. Prove DE is parallel to BC and DE=BC/2 using dilation.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Derive why a triangle's area scales by k² under a positive dilation k, rather than k.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
