Unit 7 – Scale Drawings and Dilations

Day Classwork Day Homework

Tuesday 12/4

Unit 6 Test

Wednesday 12/5

Properties of Scale Drawings

Scale Drawings Using Constructions 1 HW 7.1

Thursday 12/6

Dilations and Scale Drawings with Various

Centers of Dilation

2 HW 7.2

Friday 12/7

Computer Lab 120 (Periods 1 & 3)

Library Lab (Period 6)

Monday 12/10

Triangle Side Splitter Theorem

Midsegments of Triangles

3 HW 7.3

Tuesday 12/11

Dividing a Line Segment into Equal Segments

Unit 7 Quiz 1 4 HW 7.4

Wednesday 12/12

Dilations in the Coordinate Plane 5 HW 7.5

Thursday 12/13

Review

Unit 7 Quiz 2 6 Review Sheet

Friday 12/14

Review 7 Review Sheet

Monday 12/17

Begin Unit 8 HW 8.1 (Due Wednesday)

Tuesday 12/18

Unit 7 Test 8

SCALE DRAWINGS

The scale factor r is the ratio of any length in a scale drawing relative to its corresponding length in the

original figure. A scale factor r > 1 results in an enlargement of the original figure. A scale factor of 0 < r < 1 results in a reduction of the original figure.

Examples

1. Use construction tools to create a scale drawing of ABC with a scale factor of r =2.

Measure the length of BC and B’C’. What do you notice?

Measure the angles , , ',B C B and 'C . What do you notice?

Steps:

2. Use construction tools to create a scale drawing of DEF with a scale factor of r = 3. What properties does your scale drawing share with the original figure? Explain how you know.

3. Use construction tools to create a scale drawing of XYZ with a scale factor of 1

2r .

4. Use construction tools to create a scale drawing of PQR with a scale factor or 1

4r . What

properties do the scale drawing and the original figure share? Explain how you know.

5. EFG is provided below, and one angle of scale drawing ' ' 'E F G is also provided. Use

construction tools to complete the scale drawing so that the scale factor is 3r . What properties

do the scale drawing and original figure share? Explain how you know.

6. Triangle ABC is provided below, and one side of scale drawing ' ' 'A B C is also provided. Use construction tools to complete the scale drawing and determine the scale factor.

MORE SCALE DRAWINGS

For 𝑟>0, a dilation with center 𝑂 and scale factor 𝑟 is denoted ,O rD

For the center 𝑂, ,O rD (𝑂)=𝑂

For any other point 𝑃, ,O rD (𝑃) is the point 𝑃 on the ray OP so that 'OP r OP

1. Create a scale drawing of the figure below about center O and scale factor 1

2r .

Does A’B’C’D’E’ look like a scale drawing? How can we verify this?

2. Create a scale drawing of the figure below about center O and scale factor 2r . Verify that the

resulting figure is in fact a scale drawing by showing that corresponding side lengths are in constant proportion and the corresponding angles are equal in measurement.

3. Create a scale drawing of the figure below about center O and scale factor 3r .

4. ' ' 'A B C is a scale drawing of ABC . Use your straight edge to determine the location of the

center O used for the scale drawing.

5. Use the figure below, center O, a scale factor of 1

2r to create a scale drawing. Verify that the

resulting figure is in fact a scale drawing by showing that corresponding side lengths are in constant proportion and that the corresponding angles are equal in measurement.

TRIANGLE PROPORTIONS

Triangle Proportionality Theorem (Triangle Side Splitter Theorem) Example Figure

A line segment divides two sides of a triangle into

segments of proportional lengths if and only if it is

parallel to the third side of the triangle.

Examples In ∆𝑃𝑅𝑄, 𝑆𝑇̅̅̅̅ ∥ 𝑅𝑄̅̅ ̅̅ . 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠 𝑏𝑎𝑠𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛.

1. PT = 7.5, TQ = 3, and SR = 2.5. Find PS.

2. PS = 12.5, SR = 5, and PT = 15. Find TQ.

3. 12PS , 8ST , and 6SR . Find RQ .

4. In DEF , EH = 3, HF = 9, and DG is one third the length of GF . Is DE GH ?

A _______________________________________________ is a segment with endpoints that are the

midpoints of two sides of the triangle. Every triangle has three midsegments.

Triangle Mid-Segment Theorem

Example Figure

A midsegment of a triangle is parallel to one side of the triangle, and its length is

one half the length of that side.

Examples In the figure, XY and XZ are midsegments of RST . Find each measure below.

1. XZ

2. ST

3. 𝑚∠𝑅𝑌𝑋

4. 𝑥 = _______ 5. 𝑥 = _______

Perimeter of △ 𝐴𝐵𝐶 =_______ 𝑦 = _______

Another special case of the Triangle Proportionality Theorem involves three of more parallel lines cut by two transversals.

6. Megan is drawing a hallway in one-point perspective. She uses the guidelines

shown to draw two windows on the left wall. If segments AD , BC , WZ , and

XY are all parallel, AB = 8 cm, DC = 9 cm, and ZY = 5 cm, find WX.

7. In △ 𝑅𝑆𝑇, the midpoints of each side have been marked by points 𝑋, 𝑌, and 𝑍.

Mark the halves of each side divided by the midpoint with a congruency mark. Remember to distinguish congruency marks for each side.

Draw mid-segments 𝑋𝑌, 𝑌𝑍, and 𝑋𝑍. Mark each mid-segment with the appropriate congruency mark from the sides of the triangle.

a. What conclusion can you draw about the four triangles within △ 𝑅𝑆𝑇? Explain Why.

___________________________________________________________________________________________

b. State the appropriate correspondences between the four triangles within △ 𝑅𝑆𝑇.

___________________________________________________________________________________________

c. State a correspondence between △ 𝑅𝑆𝑇 and any one of the four small triangles.

___________________________________________________________________________________________

DIVIDING A LINE SEGMENT (DILATION METHOD)

1. Divide AB into four segments of equal lengths.

Describe your steps:

2. Use the Dilation Method to divide PQ into 9 equal-sized pieces.

3. If the segment below represents the interval from zero to one on the number line, locate and

label 4

7.

DILATIONS IN THE COORDINATE PLANE

A dilation is a transformation of an object by increasing or decreasing the object by a factor of r.

Types of Dilations Symbols Examples Figures

Enlargement

Reduction

Examples Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation.

a. b.

c. d.

1. Quadrilateral JKLM has vertices J(-2, 4), K(-2, -2), L(-4, -2),

M(-4, 2). Graph the image of JKLM after a dilation centered at the origin with a scale factor of 2.5.

2. Find the image of each polygon below with the given

vertices after a dilation centered at the origin with the scale factor.

a. Q (0, 6), R (-6, -3), S (6, -3); r = 1/3

b. A (2, 1), B (0, 3), C (-1, 2), D (0, 1); r = 2

3. A dilation with center O1 and scale factor 1/2 maps figure F onto F’. A dilation with center O2 and scale factor 1/2 maps figure F’ to F’’. Draw figures F’ and F’’, and then find the center O and scale factor r of the dilation that takes F to F’’.

4. If a figure T is dilated from center O1 with a scale factor 1

3

4r to yield image T’, and figure T’ is

then dilated from center O2 with a scale factor 2

4

3r to yield figure T’’. Explain why figure T and T’’

are congruent.