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In this lesson, we will solve problems involving scale drawings of **geometric figures**, draw (freehand, with ruler and protractor) geometric shapes with given conditions, and describe two-dimensional figures from slicing three-dimensional figures.

**See the fact file below for more information on the understanding geometric figures or alternatively, you can download our 28-page Understanding Geometric Figures worksheet pack to utilise within the classroom or home environment.**

## Key Facts & Information

### SCALE DRAWINGS

- Scale drawings are diagrams of real measurements with a different unit of measurement, arranged so as to have the same shape as the original measurement they represent.
- The scale defines the relation between the unit of measurement in the drawing and that of the original. Some examples of scale models include photographs, miniature houses, model trains, architectural designs, maps, and technical drawings for science and engineering.
- Linear dimensions on scale models are proportional to the corresponding dimension on the original. The ratio of any length in the drawing to the corresponding original length is the scale of the drawing.
- An image is a scaled copy of the original if the shape is resized in a way that does not distort it.
- An image and its scaled copy have corresponding parts, or parts that are in the same position in relation to the rest of the image. These portions could be points, segments, or angles.
- Each point in Figure 1 has a corresponding point in Figure 2. For example, vertex B corresponds to vertex H and vertex A corresponds to vertex G.
- Each segment in Figure 1 has a corresponding segment in Figure 2. For example, line segment FE corresponds to line segment LK.
- Each angle in Figure 1 has a corresponding segment in Figure 2. For example, angle EDC corresponds to angle KJI.
- The scale factor between Figure 1 and Figure 2 is 2, because all of the lengths in Figure 2 are twice the corresponding lengths in the original image. The angle measures in Figure 2 are the same as the corresponding angle measures in Figure 1.
- In creating a scaled copy, we multiply the lengths of the original image by a scale factor.
- For example, to draw a scaled copy of triangle ABC, where the base is 8 units, we will use a scale factor of 4, thus multiplying all the side lengths by 4. In triangle DEF, each side is four times the length of the corresponding side in triangle ABC.
- 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 image by the same scale factor, whether or not the endpoints are connected by a segment.
- All angles in the scaled copy have equal measurements as the corresponding angles in the original image, as in these triangles.
- These observations can give you reasons why one figure is not a scaled copy of another.
- The size of the scale factors can affect the size of the scaled copy.

### SCALING AND AREA

- Scaling also affects lengths and areas. When we create a scaled copy, all original lengths are multiplied by the scale factor. If we make a copy of a rectangle with side lengths 3 units and 6 units, by using a scale factor of 3, the side lengths of the copy will be 9 units and 18 units.
- The area of the copy, however, changes by a factor of (scale factor)². If each side length of the scaled copy is thrice the original side length, then the area of the scaled copy will be nine times the area of the original, because 3² is 9. Lengths are one-dimensional, so in a scaled copy, they vary by the scale factor. Area, on the other hand, is two-dimensional, so it differs by the square of the scale factor.
- We can see this is applicable for a rectangle with length l and width w. If we resize the rectangle by a scale factor of s, we get a rectangle with length s · l and width s · w. The area of the scaled rectangle is A = (s · l) · (s · w), so A = s² · (l · w). This is also applicable to scaled copies of other two-dimensional figures, not just for rectangles.

### CREATING SCALE DRAWINGS

- If we want to make a scale drawing of a room’s floor plan that has a scale of “1 inch to 4 feet”, we can split the actual lengths of the room (in feet) by 4 to figure out the corresponding lengths (in inches) for the scale drawing.
- Suppose the longest wall is 16 feet long. We should draw a line 4 inches long to represent this wall, since 16 / 4 is 4.
- Scales:
- 1 inch to 4 feet
- ½ inch to 2 feet
- ¼ inch to 1 foot

- The three scales above are all equivalent, since they represent the same relationship between lengths on a drawing and actual lengths.
- Any of the three could be used to find actual lengths and scaled lengths.
- The size of a scale drawing is affected by the choice of scale.

### DRAWING GEOMETRIC FIGURES WITH GIVEN CONDITIONS

- You can draw more than one kind of triangle given certain conditions.
- For example, “sides measuring 5 units and 6 units, and an angle measuring 32°” could be describe two triangles that are not identical copies of each other. Sometimes, there is only one unique triangle given a condition.
- For example, here are two identical copies of a triangle with sides of length 3 units and an angle measuring 60°. It is impossible to draw a different triangle with this condition.
- There are also cases when it is not possible to draw a triangle given certain conditions.
- For example, there is no triangle with sides measuring 4 inches, 5 inches, and 12 inches. You can try to draw it and see for yourself.
- There are also cases when it is not possible to draw a triangle given certain conditions.

### SLICING THREE-DIMENSIONAL SHAPES

- A cross-section is a two-dimensional shape produced from cutting a three-dimensional shape with a plane. The shape of the cross-section depends on the type of “cut” (vertical, horizontal, angled).
- A vertical cut means you are cutting up and down. A horizontal cut, on the other hand, would be cutting from side to side.

### SLICING A RECTANGULAR PRISM

- A vertical slice can be parallel to the left and right faces. The cross section always has the same shape and dimensions as these faces. A vertical slice can also be parallel to the front and back faces. The cross section always has identical shapes and dimensions as these faces.
- A horizontal slice is parallel to the bases. The cross section always has identical shapes and dimensions as these faces.

### SLICING A RECTANGULAR PYRAMID

- If you make any horizontal slice of a rectangular pyramid, the resulting cross section is a rectangle. The size of the rectangle depends on the distance of the slice from the base.
- If you make a vertical slice of a rectangular pyramid through the vertex, the resulting cross section is an isosceles triangle. The base of the triangle is equal in length to an edge of the triangular base. The height of the triangle is equal to the height of the pyramid.

**Understanding Geometric Figures Worksheets**

This is a fantastic bundle which includes everything you need to know about the understanding geometric figures across 28 in-depth pages. These are** ready-to-use Understanding Geometric Figures worksheets that are perfect for teaching students how to solve problems involving scale drawings of geometric figures, draw (freehand, with ruler and protractor) geometric shapes with given conditions, and describe two-dimensional figures from slicing three-dimensional figures. **

### Complete List Of Included Worksheets

- Lesson Plan
- Understanding Geometric Figures
- More or Less?
- Drawing Angles
- Shapes and Angles
- Simon Says
- Testing Triangles
- Slicing 3D Shapes
- Scaled Rectangles
- Scaled Copy
- Scaled Angle Names
- Bedroom Floor Plan

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Link will appear as Understanding Geometric Figures Facts & Worksheets: https://kidskonnect.com - KidsKonnect, July 30, 2020

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