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Solutions for Session 6, Part C
See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6| C7
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Problem C1 | |
a. | All sketches produce a similar triangle:
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b. | No, it is four times larger. In all cases, the area of a doubled polygon is four times the area of the original. Tripling the side lengths results in a polygon nine times larger in area. Quadrupling the side lengths results in a polygon 16 times larger in area -- and so on. |
<< back to Problem C1
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Problem C2 | |
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If Ao is the area of the original polygon, then we can write the following:
<< back to Problem C2
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Problem C3 | |
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The number of copies needed is the square of the scale factor. For example, making a copy that is three times larger in each direction will take nine copies of the original shape.
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Problem C4 | |
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The area of the enlarged figure is the original area multiplied by the square of the scale factor.
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Problem C5 | |
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Because the scale factor is 3, the area is nine times larger. Therefore, the area of the enlarged figure is 72 cm2.
For example, suppose the original figure were a 4-by-2 rectangle (with an area of 8 cm2). The new shape would then be 12 by 6, with an area of 72 cm2 -- nine times the original area. Here's how it breaks down:
A = 12 • 6
A = (4 • 3) • (2 • 3)
A = 4 • (3 • 2) • 3associative property
A = 4 • (2 • 3) • 3commutative property
A = (4 • 2) • (3 • 3) associative property
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Problem C6 | |
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One way to think about it is that enlarging an object will require k copies of that object in each direction: k copies in one direction, multiplied by k copies in the other direction, for a total of k2.
<< back to Problem C6
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Problem C7 | |
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All of these polygons are rep-tiles. Most rep-tiles have side lengths that have a common factor, but this is not a requirement.
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