Unit 2, Lesson 3
Scaled Relationships
Accelerated 7
3.1
Warm-up
What do you notice? What do you wonder?
Sign in to view assessments and invite other educators
Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.
What do you notice? What do you wonder?
Measure at least one set of corresponding angles using a protractor. Record your measurements to the nearest .
What do you notice about the angle measures?
Pause here so your teacher can review your work.
The side lengths of the polygons are hard to tell from the grid, but there are other corresponding distances that are easier to compare. Identify the distances in the other two polygons that correspond to and , and record them in the table.
| quadrilateral | distance that corresponds to |
distance that corresponds to |
|---|---|---|
Look at the values in the table. What do you notice?
Are these three quadrilaterals scaled copies? Explain your reasoning.
Your teacher will give you one of the six pieces of a puzzle.
When a figure is a scaled copy of another figure, we know that:
All distances in the copy can be found by multiplying the corresponding distances in the original figure by the same scale factor, whether or not the endpoints are connected by a segment.
For example, Polygon is a scaled copy of Polygon . The scale factor is 3. The distance from to is 6, which is three times the distance from to .
Polygon ABCDEF and its scaled copy Polygon STUVWX. The vertices of Polygon ABCDEF starting at A going counterclockwise are as follows. Vertex B is 1 unit to the left and 2 units down. Vertex C is 2 units down. Vertex D is 1 unit up and 1 unit to the right. Vertex E is 1 unit down and 1 unit to the right. Vertex F is 2 units up. The vertices of Polygon STUVWX starting at S going counterclockwise are as follows. Vertex T is 3 units to the left and 6 units down. Vertex U is 6 units down. Vertex V is 3 units up and 3 units to the right. Vertex W is 3 units down and 3 units to the right. Vertex X is 6 units up. 1 unit=1 square on the grid.
These observations can help explain why one figure is not a scaled copy of another.
For example, the second rectangle is not a scaled copy of the first rectangle, even though their corresponding angles have the same measure. Different pairs of corresponding lengths have different scale factors, but .
When one figure is a scaled copy of another, the size of the scale factor affects the size of the copy.
When a figure is scaled by a scale factor greater than 1, the copy is larger than the original. When the scale factor is less than 1, the copy is smaller. When the scale factor is exactly 1, the copy is the same size as the original.
Triangle is a larger scaled copy of triangle , because the scale factor from to is . Triangle is a smaller scaled copy of triangle , because the scale factor from to is .
Two triangles; one labeled A B C with horizontal A B and the other D E F with horizontal D E. The length of A B is labeled 4. The length of B C is labeled 3. The length of C A is labeled 5. The length of D E is labeled 6. The length of E F is labeled 4.5. The length of F D is labeled 7.5. An arrow from triangle A B C pointing to triangle D E F is labeled, times 3 halves. An arrow from triangle D E F pointing to triangle A B C is labeled times 2 thirds.
This means that triangles and are scaled copies of each other. It also shows that scaling can be reversed using reciprocal scale factors, such as and .
In other words, if we scale Figure A using a scale factor of 4 to create Figure B, we can scale Figure B using the reciprocal scale factor, , to create Figure A.
Two numbers that multiply to equal 1 are reciprocals.