Unit 1, Lesson 4
Scaled Relationships
Grade 7
4.1
Warm-up
Notice and Wonder: Three Quadrilaterals
What do you notice? What do you wonder?
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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
| 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.
Here are two quadrilaterals.
Two quadrilaterals on a coordinate plane. The first figure is labeled JXNY. Point X is 2 units to the left and 8 units up from point J. Point N is 2 units to the right and 1 unit up from point X. Point Y is 4 units to the right and 1 unit down from point N. Point J is 4 units to the left and 8 units down from point Y. Point N is directly above point J. The second figure is labeled ZHCS. Point Z is 1 unit to the left and 5 units up from point S. Point H is 1 unit to the right and 1 unit up from point Z. Point C is 3 units to the right and 1 unit down from point H. Point S is 3 units to the left and 5 units down from point C. Point H is directly above point S.
| quadrilateral | horizontal distance | vertical distance |
|---|---|---|
Here are two more quadrilaterals.
The angle measures, in degrees, for both trapezoids are: 60, 60, 120, 120. In A, B, C, D, the top length is 2, bottom length is 6, both sides lengths are 4. In E, F, G, H, the top length is 1, bottom length is 4 and both side lengths are 3.
Here are two pictures of a bird. Find evidence that one picture is not a scaled copy of the other. Be prepared to explain your reasoning.
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
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,