Dilations
Is the following an enlargement or reduction?
K= 3/2
Enlargement
There is a pair of similar triangles. The length of the first triangle's sides are 12, 18, and 36 inches. The length of the second triangle's sides are 2, 3, and 6 inches. Give the scale ratio of the first triangle to the second triangle.
6:1
Out of the three ways to prove a triangle is similar, which way is being described?
If all corresponding sides are proportional, then the triangles are similar.
SSS~
Recall: To dilate a figure, you multiply the coordinates by a ________.
Scale factor
Given the point and its image after a dilation, determine the scale factor.
X(2, 5) ----> X' (3, 7.5)
k=?
1.5
There is a pair of similar triangles. The length of the first triangle's sides are 15, 18, and 21 inches. The length of the second triangle's sides are 5, 6, and 7 inches. Give the perimeter ratio of the first triangle to the second triangle.
3:1
Out of the three ways to prove a triangle is similar, which way is being described?
If two corresponding angles are congruent, then the triangles are similar.
AA~
Recall: The ratio of sizes of two similar figures is called ________.
Similarity ratio
Find the a triangle's coordinates after a dilation with the given scale factor.
A(-6, -8), B(5, 1), C(0, -2) k = 3
A'(-18, -24), B'(15,3), C'(0, -6)
The images on Erica's digital camera have a width-to-length ratio of 2:3. She wants to make an 8 inches by 10 inches print of one of her photographs.
Is this possible? If so explain why. If not, give the ratio needed to make an 8 inches by 10 inches print of one of her photographs.
No it is not possible. The correct ratio to make an 8 inches by 10 inches print is 4:5.
Out of the three ways to prove a triangle is similar, which way is being described?
If two corresponding sides are proportional and the included angles are congruent, then the triangles are similar.
SAS~
Recall: What are three different ways to prove that two triangles are similar?
SSS, SAS, AA
P(-2, 2), Q(4, 2), R(2, -6), S(-4, -6). Dilate with respect to the origin using a scale factor of 5. Then, dilate with respect to the origin using a scale factor of 0.5.
P''(-5, 5), Q''(10, 5), R''(5, -15), S''(-10, -15)
Or was wondering how tall a tree is in her backyard. She grabbed a mirror and placed it on the ground 42 feet away from the base of the tree. Or then walked backwards until she was able to see the top of the tree in the mirror. If Or's eyes are 6 feet off the ground and she is standing 16 feet away from the mirror, how tall is the tree?
15.75 feet
Are triangles ABC and DEF are similar by AA~?
<A=75 degrees, <B=74 degrees, <C=? degrees
<D=75 degrees, <E=? degrees, <F=31 degrees
Yes or no? Explain why or why not.
Yes, since two angles in triangle ABC are congruent to the corresponding two angles in triangle DEF, the triangles are similar by the AA~ Postulate.
Recall: What word is used when k=1?
Isometry
F(-9, -9), G(-3, -6), H(-3, -9). Dilate with respect to the origin using a scale factor of 2/3. Then translate 6 units up.
F''(-6, 0), G''(-2, 2), H''(-2, 0)
The population density of the city of Wellington is 12,000 people per square mile. The area of Wellington is 12. The neighboring city Morrison is a dilation of Wellington with scale factor 1.2. If Morrison has 3/4 the population of Wellington, what is the population density of Morrison?
Hint: Population density: Population/Area
6250 people per square mile
Are triangles LMN and OPQ are similar by AA~?
<L=35 degrees, <M=65 degrees, <N=? degrees
<O=? degrees, <P=34 degrees, <Q=80 degrees
Yes or no? Explain why or why not.
No, <L is not congruent to <O, and <M is not congruent to <Q, so the triangles are not similar by the AA~ postulate.
What is being described below?
An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.
Angle bisector theorem