Unit 6: Similarity and Dilation
Scaling a shape up or down without distorting it, the shortcuts for proving two triangles are similar, and what scaling secretly does to area.
7 topics · 25 questions
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Dilations and Scale Factor
- How it worksHow does a center of dilation and a scale factor together stretch every point away from or toward one spot?
- Worked exampleFind the image of the point (3, 4) dilated by a scale factor of 2 centered at the origin.
- Common mistakeWhy isn't a dilation with a scale factor less than 1 actually shrinking the shape's angles?
- Going furtherWhat happens to a figure when it's dilated by a negative scale factor?
Your turn1 of 2
Point (4, 3) is dilated by a scale factor of 5, centered at the origin. Find the sum of the coordinates of the image.Similar Polygons
- How it worksHow do corresponding angles staying equal and corresponding sides staying proportional together define similarity?
- Worked exampleFind the missing side of a similar rectangle if a 4 by 6 rectangle is similar to one with a side of 10.
- Common mistakeWhy isn't having all four sides proportional enough to guarantee two quadrilaterals are similar?
- Going furtherWhat happens to the perimeter ratio of two similar polygons compared to their side length ratio?
Your turn1 of 2
A rectangle has a short side of 5 and a long side of 16. It is similar to a second rectangle whose corresponding short side measures 14. Find the second rectangle's corresponding long side.Triangle Similarity Criteria: AA, SAS, SSS
- How it worksHow does the angle-angle criterion skip checking any side length at all to prove similarity?
- Worked exampleShow that two triangles are similar by SAS given sides of 4 and 6 with a 50 degree included angle, matched to sides of 8 and 12 with a 50 degree included angle.
- Common mistakeWhy doesn't SSA prove similarity any more reliably than it proves congruence?
Your turn1 of 2
One triangle has two sides measuring 8 and 8, with an included angle of 159 degrees. A second triangle has corresponding sides of 3 times 8 and 3 times 8 with the same included angle, confirming SAS similarity with scale factor 3. Find the length of the second triangle's third side.Parallel Lines Cutting Proportional Sides
- How it worksHow does the side-splitter theorem prove the two smaller triangles formed are similar to the original?
- Worked exampleFind the missing length when a line parallel to a triangle's base splits one side into segments of 4 and 6, with the other side's first segment equal to 5.
- Going furtherWhat happens to the proportional side lengths when three or more parallel lines cut across two transversals?
Your turn1 of 2
A line parallel to a triangle's base splits one side into segments of 8 and 7, measured from the apex down. It splits the other side's first segment, also measured from the apex, into a length of 14. Find the length of that other side's remaining segment.Indirect Measurement Using Similar Triangles
- How it worksHow do a tree and its shadow form one triangle similar to a much smaller person and their shadow?
- Worked exampleFind the height of a flagpole that casts a 24 foot shadow at the same time a 5 foot person casts a 4 foot shadow.
- Common mistakeWhy does indirect measurement with shadows only work when both shadows are measured at the same moment?
- Going furtherWhat happens to the indirect-measurement setup when a mirror is used on the ground instead of shadows?
Your turn1 of 2
A person 6 feet tall casts a shadow 7 feet long at the same moment a flagpole casts a shadow 57 feet long. Find the height of the flagpole.Proving the Pythagorean Theorem with Similar Triangles
- How it worksHow does the altitude to the hypotenuse split a right triangle into two smaller triangles similar to the original?
- Worked exampleFind the length of the altitude to the hypotenuse in a right triangle with hypotenuse segments of 4 and 9.
- Ties togetherHow does this similarity-based proof reach the exact same a squared plus b squared equals c squared as the area-rearrangement proofs?
Your turn1 of 2
In a right triangle, the altitude to the hypotenuse splits the hypotenuse into segments of 15 and 14. Find the length of the altitude.How Scale Factor Affects Area
- How it worksHow does squaring the scale factor account for area growing in two directions at once?
- Worked exampleFind the area of a triangle scaled by a factor of 3 if the original triangle's area is 12.
- Common mistakeWhy doesn't scaling a shape's sides by a factor of 2 also just double its area?
- Going furtherWhat happens to a solid's volume, instead of its area, when every one of its lengths is scaled by the same factor?
Your turn1 of 2
A shape has an area of 35. Find the area of the shape after it is scaled by a factor of 5.