unit 7 scale drawings and dilations -...
TRANSCRIPT
Unit 7 β Scale Drawings and Dilations
Day Classwork Day Homework
Friday 12/1
Unit 6 Test
Monday 12/4
Properties of Scale Drawings
Scale Drawings Using Constructions 1 HW 7.1
Tuesday 12/5
Dilations and Scale Drawings with Various
Centers of Dilation
2 HW 7.2
Wednesday 12/6
Triangle Side Splitter Theorem
Midsegments of Triangles
Unit 7 Quiz 1
3 HW 7.3
Thursday 12/7
Dividing a Line Segment into Equal Segments 4 HW 7.4
Friday 12/8
Dilations in the Coordinate Plane 5 HW 7.5
Monday 12/11
Review
Unit 7 Quiz 2 6 Review Sheet
Tuesday 12/12
Review 7 Review Sheet
Wednesday 12/13
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 BβCβ. 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 π>0, a dilation with center π and scale factor π is denoted ,O rD
For the center π, ,O rD (π)=π
For any other point π, ,O rD (π) is the point π 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 AβBβCβDβEβ 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.
DILATIONS
Dilation Theorem
Example Figure
If a dilation with center O and scale factor r sends point π to πβ² and π to
πβ², then |πβ²πβ²|=π|PQ|. Furthermore, if πβ 1 and π,, and Q are the vertices of
a triangle, then 'Q'PQ P .
Examples:
1. Given the diagram below, determine if DEF is a scale drawing of DGH . Explain why or why
not.
2. Two different points R and Y are dilated from S with a scale factor of 3
4, and RY = 15. Use the
Dilation Theorem to describe two facts that are known about RβYβ.
Summary
a. Which transformations of the plane are distance-preserving transformations? Provide an example
of what this property means.
b. Which transformations of the plane preserve angle measure? Provide one example of what this
property means.
c. Which transformation is not considered a rigid motion and why?
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 βππ π, ππΜ Μ Μ Μ β₯ π πΜ Μ Μ Μ . πΉπππ π‘βπ ππππππ€πππ ππππ π’πππ πππ ππ ππ π‘βπ πππ£ππ ππππππππ‘πππ.
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. πβ π ππ
4. π₯ = _______ 5. π₯ = _______
Perimeter of β³ π΄π΅πΆ =_______ π¦ = _______
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 β³ π ππ, the midpoints of each side have been marked by points π, π, and π.
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 ππ, ππ, and ππ. 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 β³ π ππ? Explain Why.
___________________________________________________________________________________________
b. State the appropriate correspondences between the four triangles within β³ π ππ.
___________________________________________________________________________________________
c. State a correspondence between β³ π ππ 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 Fβ. A dilation with center O2 and scale factor 1/2 maps figure Fβ to Fββ. Draw figures Fβ and Fββ, and then find the center O and scale factor r of the dilation that takes F to Fββ.
4. If a figure T is dilated from center O1 with a scale factor 1
3
4r to yield image Tβ, and figure Tβ is
then dilated from center O2 with a scale factor 2
4
3r to yield figure Tββ. Explain why figure T and Tββ
are congruent.