Download 1. If this is a dilation, there is a point of dilation. Is this a dilation

Honors Geometry Mid-Term Exam Study Guide Part 1 (Wed 10/1/14) Name_____________________________
MCC9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor.
1. If this is a dilation, there is a point of dilation. Is this a dilation? _____ If so, locate the point of
dilation and give the coordinates of this reference point. ( ____ , ____).
2. The scale factor from the small triangle to the large triangle is _____ but the scale factor from
the large triangle to the small triangle is _____.
3. We can show that these triangles are similar figures because the 3 corresponding sides are
proportional (SSS ~). Examine a single side, draw the line segments to the point of dilation. Can
you see this is an application of the side-splitting theorem for triangles. What is true about
these corresponding sides (shown as segments in figure 1 below) when the point of dilation is
not on either segment?
a) They segments are proportional by a scale factor (constant of proportionality) of
either 3 or
b) The segments are parallel.
c) The corresponding angles for the transversals between the sides are congruent.
d) Al of the above.
4. Dilations from the origin are easier! Dilate the given segment from the origin by a factor of 2.
Length of original segment?____
Length of original segment?____
Length of dilated segment? ____
Length of dilated segment? ____
Was the dilated segment parallel to the original? _____ Was the dilated segment parallel to the original? _____
If the point of dilation IS NOT on the line segment then the original segment and the dilated segment are ___________.
If the point of dilation IS on the line segment then the original segment and the dilated segment are ________________.-
5. a) If this is a dilation then locate the point of dilation. Label it as “A.”
b) If the dilation scale factor from the smaller hexagon to larger one is 2,
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.
How are the sides related? ___________________________
How are the angles related? ___________________________
How are the perimeters related? __________________________
How are the areas related? ______________________________
6. Are these similar figures?___ How can you tell? _______________ The scale factor from ABCD to A’B’C’D’ is ___
MCC9-12.G.SRT.2 Meaning of Similar Figures
7. If these hexagons are similar ( ABCDEF ~ TUVQES ), then …
A) If AF = 10, UT = 21, and ST = 15 then the scale factor is ____ and AB = ___
B) What information does THE DEFINITION OF Similar Figures tell us about the corresponding angles?
____________________________________________________________________________________
C) What information does THE DEFINITION OF Similar Figures tell us about the corresponding sides?
____________________________________________________________________________________
8. a)If we want to prove that ∆MKD ~∆LPG using only the definition of similar triangles we must show 6 things:
/ M ~ ______ , / K ~ ______, / D ~ ______,
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Luckily, we have short-cuts for proving similarity… why are these triangles similar?
_____________________
b)
___________________
c)
_____________________
d)
MCC9-12.G.SRT.3 Similar Triangles
9. Given:
in the figure above. A,C, and E are collinear and B,C,D are collinear.
Prove:
ǁ
and A,C,E collinear and B,C,D collinear Reason?
/ BCA and / DCE are vertical angles
/ BCA ~ / DCE
Definition of Vertical Angles (non-adjacent angles
formed by 2 intersecting lines)
Reason?
∆BCA ~ __________
Reason?
/ A ~ _____
Reason?
Statement?
If the alternate interior angles are congruent then the
lines are parallel.
10. Given: / A ~ / E. We also are given that A,C, and E are collinear and B,C,D are collinear.
Prove: ∆BCA
~ ∆DCE
Paragraph Proof: We are given that these are intersecting lines so ______ and _______ are vertical
angles (by definition). This means these angles are congruent. Why? ________________________. We
are given that / A ~ / E. We now have enough information to conclude that ∆BCA ~ ∆DCE
because of the ________________ theorem.
11. Are these triangles congruent? ____________ Why/Why not?________________________
Are they similar? ________ ? Why/why not? ____________________
If BC = 6, CA = 8, AB = 7, and AE = 20 then the perimeter of ∆ECD is _______ units.
MCC9-12.G.SRT.4 Prove theorems about triangles.
Figure showing lengths of segments
Figure with Vertices of ∆’s labeled
12. If you construct a line parallel to any side of a triangle then you get some interesting theorems.
a) Prove the smaller triangle is similar to the larger.
Since the lines are parallel, the ___________________ angles are congruent ( / BRT ~ / BAC and / BTR ~ / BCA)
Since 2 pair of corresponding angles are congruent, ∆RBT ~ ∆ABC. Why? ___________________________________
Now that we know the triangles are similar, write a proportionality statement for the sides of the
triangles.
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b) Continue the line of proof from part “a” to Prove the sides that were split by the parallel construction
are split proportionally (often called the Side-Splitter Theorem)
13. Prove the Pythagorean Theorem
14. Prove: If you have an isosceles triangle, the angles opposite the congruent sides are congruent.
MCC9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems
15. x = _______ , AB = _____, AC = _______
16. if you look at this figure you can see a dilation of
this as a dilation of
from point _____ by a scale factor of _____. You can also view
from point _____ by a scale factor of _____.
17. Shadow Similarity: If the sun’s rays are parallel to each other then the ground acts as a transversal between
the rays as shown.
Mark the angles that are congruent. Why is this true?
Draw the triangles (assume │ to ground).
Why are they similar?
If the shadow of the giraffe is 18 ft. long, the giraffe is 12 feet tall and the shadow of the tree is 20 feet, how tall is
the tree? __________ ft
Which equation would NOT work?
#18. x = ____ and y = ______
a)
b)
c)
d)
#19. Are these triangles similar? ___ why? _________________
x = ________ y = _____
#20.
If the top angle measure is 50o then the two base angles each measure _____o.