Honors Geometry B Unit 3: Similarity
Are distances and/or angles preserved in a dilation?
Only angles are preserved in a dilation.
post-image
the image produced after a transformation
image
the new figure after a transformation
preimage
the original figure before a transformation
pre-image
the original shape
Line segment A′B′¯¯¯¯¯¯¯¯¯¯A′B′¯, whose endpoints are (4, −2)(4, −2) and (−2, −6)(−2, −6), is the image of AB¯¯¯¯¯¯¯¯AB¯ after a dilation of 1212 centered at the origin. What are the coordinates of the endpoints of AB¯¯¯¯¯¯¯¯AB¯?
(8, −4) and (−4, −12)
post-dilated point
any point that lies on the transformed image
Which transformations preserve angle measures and side lengths? Which transformations do not preserve side lengths and may not preserve angle measures?
A rigid transformation, such as a translation, rotation, or reflection, preserves angle measures and side lengths. A nonrigid transformation, such as a dilation or shear transformation, does not preserve side lengths and may not preserve angle measures.
Define the terms translation, rotation, reflection, and dilation.
A translation moves a shape in one direction. A rotation is a circular movement that moves a shape around a central point by a specified angle. A reflection creates a mirror reflection of a shape about a given line. A dilation is a transformation that enlarges or reduces a shape by multiplying the lengths of all sides of a figure by a given value.
A line segment with endpoints A(−2, 1)A(−2, 1) and B(2, 1)B(2, 1) is dilated with a dilation centered at the origin and a scale factor of 5252 to create A′B′¯¯¯¯¯¯¯¯¯¯A′B′¯. Which of the following statements regarding AB¯¯¯¯¯¯¯¯AB¯ is not true?
AB=52A′B′AB=52A′B′.
If AB⎯⎯⎯⎯⎯AB¯ is dilated using a scale factor of 44 to create A‵B‵⎯⎯⎯⎯⎯⎯⎯A`B`¯, then what is the value of ABA‵B‵ABA`B`?
ABA‵B‵=14ABA`B`=14 The answer is not 44 because the scale factor is equal to the dilated length divided by the pre-dilated length, while this problem asks about the inverse.
If a triangle undergoes a similarity transformation, then what equations can be used to relate the triangles △ABC and △A`B`C`?
A`B`AB=B`C`BC=C`A`CAA`B`AB=B`C`BC=C`A`CA, ∠A≃∠A`∠A≃∠A`, ∠B≃∠B`∠B≃∠B`, and ∠C≃∠C`
Are distances and/or angles preserved in a rotation?
Both distances and angles are preserved in a rotation.
What information needs to be given to prove △ACB~△ECD by AA, SSS, and SAS?
For AA similarity, you need two angles to be congruent. ∠ACB∠ACB and ∠ECD∠ECD are vertical angles and, therefore, are congruent. You need to know one other set of congruent angles, either ∠A≅∠E∠A≅∠E or ∠B≅∠D∠B≅∠D. For SSS similarity, you need to know that all three corresponding sides are proportional, ACEC=BCDC=ABEDACEC=BCDC=ABED. For SAS similarity, you need to know that two corresponding sides are proportional and the included angles between these sides are congruent. You know that ∠ACB≅∠ECD∠ACB≅∠ECD because they are vertical angles, so you need to know that ACEC=BCDCACEC=BCDC.
SAS~ Theorem
If an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two angles are proportional, then the triangles are similar.
Why is there an AA similarity criterion but not an AA congruency criterion for triangles?
If corresponding angles are congruent, then the triangles have a similar shape but the size can vary.
SSS~ Theorem
If the corresponding sides of two triangles are proportional, then the triangles are similar.
If △ABC is similar to △XYZ, then what are the relationships between the sides and angles?
If the triangles are similar then the corresponding angles are congruent, ∠A≅∠X, ∠B≅∠Y, ∠C≅∠Z, and the corresponding sides are proportional, ABXY=BCYZ=ACXZ.
AA~ Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Jonathon says that k=AMMB=ANNCk=AMMB=ANNC, if kk is the scale factor between △ABC△ABC and △AMN△AMN. Is he correct?
Jonathon is incorrect. There is a proportional relationship given by AMMB=ANNCAMMB=ANNC, however the scale factor is equal to k=AMAB=ANACk=AMAB=ANAC.
Are two isosceles triangles always similar? Explain.
No, because the ratios of the sides that form the vertex angles are the same, but the vertex angles may not be congruent.
Is dilation a rigid transformation? Explain.
No, dilation is not a rigid transformation because it changes the lengths of the sides.
If a transformation causes the angle measures of a figure to change are the figures similar? Why or why not?
No, similar figures need to have corresponding sides proportional and corresponding angles congruent.
Given △ABC△ABC and △XYZ△XYZ: ∠B=46°∠B=46°, ∠C=79°∠C=79°, ∠X=79°∠X=79°, and ∠Y=47°∠Y=47°. Are the triangles similar? If so, write a similarity statement.
No, the triangles are not similar. △ABC△ABC has angles measures of 55°55°, 46°46°, and 79°79° respectively. △XYZ△XYZ has angle measures of 79°79°, 47°47°, and 54°54° respectively. Since the triangles don't have congruent angles, they are not similar.
Sam says that the post-dilated edge of a triangle is not necessarily collinear with the pre-dilated edge, even if the center of dilation lies on an edge, since the end points will be moved. Is he correct?
No. If a dilated line must be collinear, then any of its line segments must also be collinear. The end points will move along the same line so the edges are collinear.
Name three similarity criteria for triangles.
SSS (side-side-side), AA (angle-angle), SAS (side-angle-side)
Sarah says that △BCA~△BXY△BCA~△BXY if YY lies on AB⎯⎯⎯⎯⎯AB¯, XX lies on BC⎯⎯⎯⎯⎯BC¯, and AC←→∥XY←→AC↔∥XY↔. Is she correct? Explain.
Sarah is correct. Since AC←→∥XY←→AC↔∥XY↔, the triangles are similar by the AA criterion.
Every dilated point lies on a line formed by what two points?
The center of dilation and the pre-dilated point.
Give a center of dilation and a scale factor that dilates the line y=1y=1 and results in a line above the given line.
The center of dilation is (0,0)(0,0) and the scale factor is 22.
The Triangle Proportionality Theorem states that if a triangle is intersected by a transveral parallel to one side of the triangle, then the sides it intersects are split proportionally. The statement ∠BAC≅∠BDE is true if DE∥AC. What property allows you to state this?
The corresponding angles of a transversal crossing parallel lines are congruent.
Find the missing side lengths of the similar polygons.
The dilation scale factor is GHCD=164=4GHCD=164=4. To find the missing side lengths, multiply by the scale factor to find corresponding side lengths for the larger figure or divide by the scale factor to find corresponding side lengths for the smaller figure. BC=FG4=604=15EH=4(AD)=4(8)=32
A line segment connects points at the following coordinates (1, 2) and (3, 0). Use the origin as the center of dilation and compare the resulting line segments dilated using scale factors of 2 and 5.
The endpoints for the segment dilated by a scale factor of 2 are (2, 4) and (6, 0). This segment is twice the length of the original segment. The endpoints for the segment dilated by a scale factor of 5 are (5, 10) and (15, 0). This segment is 5 times the length of the original segment.
A line segment has a length of 16 units. If the line segment is dilated by a scale factor of 1/4, what is the length of the resulting line segment?
The length of the segment after the dilation is 4 units because 16 · 1/4 = 4
A line segment has a length of 88 units. If the line segment is dilated by a scale factor of 3/4, what is the length of the resulting line segment?
The length of the segment after the dilation is 6 units.
A line segment is dilated by a scale factor of 2 centered at a point not on the line segment. Which statement regarding the relationship between the given line segment and its image is true?
The line segments are parallel, and the image is twice the length of the given line segment.
Find a point on the line y = 2x and dilate it using a scale factor of 3, where the origin is the center of dilation. Is the dilated point on the given line?
The point (2, 4) is on the given line. When (2, 4) is dilated by a scale factor of 3, its dilated point is (6, 12). If you substitute x = 6 into the equation y = 2x, the result is 1212. Thus, the dilated point (6, 12) is also on the given line.
If the perpendicular distance from the center of dilation to a line is 66, and the distance to a parallel line is 1515, what scale factor should be used to map the first line onto the second?
The scale factor is the distance of the mapped image over the distance of the original image, 156=52=2.5156=52=2.5.
If △ABC has side lengths of 3,4, and 5, then what will the side lengths be if it is dilated by a scale factor of 3?
The side lengths of the dilated triangle will be 9, 12, and 15.
△ABC△ABC and △DEF△DEF are similar, where AB⎯⎯⎯⎯⎯AB¯ corresponds to DE⎯⎯⎯⎯⎯DE¯, AC⎯⎯⎯⎯⎯AC¯ corresponds to DF⎯⎯⎯⎯⎯DF¯, and BC⎯⎯⎯⎯⎯BC¯ corresponds to EF⎯⎯⎯⎯⎯EF¯. If m∠A=45°m∠A=45°, m∠B=45°m∠B=45°, AC=3AC=3, AB=32‾√AB=32, DF=33DF=33, find the missing angle measures and lengths of the triangles.
The sum of the angles of a triangle is 180°180°, so, m∠C=90°m∠C=90°. Since the triangles are similar, the corresponding angles are congruent. Thus, m∠D=45°m∠D=45°, m∠E=45°m∠E=45°, and m∠F=90°m∠F=90°. BC=3BC=3, as a result of using the Pythagorean Theorem. Since the triangles are similar, the corresponding side lengths are proportional. So, ABDE=ACDF=BCEFABDE=ACDF=BCEF. We can use these proportions to find the side lengths of EF⎯⎯⎯⎯⎯EF¯ and DE⎯⎯⎯⎯⎯DE¯. ABDE32√DE33·32‾√332‾√====ACDF3333DEDEABDE=ACDF32DE=33333·32=3DE332=DE ACDF3333EFEF====BCEF3EF3·3333ACDF=BCEF333=3EF3EF=3·33EF=33 Thus, m∠C=90°m∠C=90°, m∠D=45°m∠D=45°, m∠E=45°m∠E=45°, m∠F=90°m∠F=90°, BC=3BC=3, DE=332‾√DE=332, and EF=33EF=33.
Explain why the triangles are similar. Then find the distance represented by x.
There is a pair of congruent vertical angles and a pair of congruent right angles, so the triangles are similar by the AA similarity criterion. The distance represented by x is 200 feet.
What change is there in a line passing through the center of the dilation after dilation?
There is no change as the dilated line lies on the original line.
Are the two triangles similar? Explain.
There is not enough information given to determine if the triangles are similar by the AA similarity criteria.
non-rigid transformation
Transformation like a dilation that does not maintain size and shape.
What scale factor should be used to dilate a line segment of length 2 so that the resulting length is 8?
Use a scale factor of 4. The new length divided by the old length is the scale factor, 8/2=4
Suppose △ABC has angle measures of 15° and 60°, and △EFG has angle measures of 60° and 105°. Are the two triangles similar? Why or why not?
Yes, the triangles are similar by the AA similarity criterion. The angle measures of a triangle add up to 180°, thus the third angle measure of △EFG is 180°−60°−105°=15°. Since △ABCand △EFG have two congruent angles, then the triangles are similar by the AA similarity criterion.
Can one triangle be mapped to the other using only similarity transformations?
Yes. △ABC is translated so B is mapped to E, then △ABC is rotated so BA and EF are aligned, finally △ABC is dilated so BA and EF are congruent.
reduction
a dilation process that produces an image smaller than its original size
shear transformation
a transformation in which all points of a preimage along a given line remain fixed while other points are shifted parallel to the line by a distance proportional to their perpendicular distance from the line
rigid transformation
a transformation in which the image is congruent to the preimage; also called an isometry
nonrigid transformation
a transformation in which the image is not congruent to the preimage
dilation
a transformation in which the image is the same shape but a different size than the preimage
rotation
a transformation in which the preimage is turned about a fixed point to produce a new image
rigid motion
a transformation that does not change the size or shape of a figure, such as translation, rotation, and reflection
non-rigid motion transformation
a transformation that does not preserve angles and/or distance in a figure
translation
a transformation that moves every point of the preimage the same distance and in the same direction
rigid-motion transformation
a transformation that preserves angles and distance, producing congruent figures
similarity transformation
a transformation that preserves angles in a figure
isometry
a transformation that preserves length and angle measure
reflection
a transformation that uses a line of reflection to create a mirror image of the preimage
pre-dilated point
any point on the pre-image
included angle
an angle between two sides
proportion
an equation stating two ratios are equal
auxiliary line
an extra line needed to complete a proof
transformation
an operation that changes or moves a preimage to produce a new image
corresponding
angles or line segments that appear in the same relative position in two similar figures
center of dilation
any fixed point on the coordinate plane about which all the points of an object are expanded or contracted by a scale factor
enlargement
dilation process that produces an image larger than its original size
Triangle Proportionality Theorem
if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally; also known as the Side-Splitter Theorem
Side-Angle-Side Similarity Theorem
if an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two angles are proportional, then the triangles are similar
Segment Addition Postulate
if points A, B, and C are collinear and point B is between A and C, then AB+BC=AC
SSS (Side-Side-Side) Similarity Theorem:
if the corresponding side lengths are proportional
AA (Angle-Angle) Similarity Postulate:
if two corresponding angles are congruent,
SAS (Side-Angle-Side) Similarity Theorem
if two corresponding sides of two triangles are proportional, and the angles between the sides are congruent.
collinear
points that lie on a single straight line
postulates
statements that are accepted as true without proof
scale factor
the constant ratio between the transformed size and the original size of the object
mapping
the process of drawing an image from a transformation
scale factors
the ratios by which images are increased or decreased in size
proportional
to be the same as or corresponding to
translate
to move or "slide" a figure in any direction
similarity transformations
transformations in which the pre-image and the new image are similar. They include rigid transformations (translation, rotation, and reflection) and non-rigid transformations (dilation).
similar triangles
triangles that have congruent corresponding angles and proportional corresponding sides
congruent triangles
triangles that have the same angle and distance measurements
congruent
two figures are congruent if they are exactly the same shape and size
indirect measurement
using proportions to find an unknown length or distance using similar figures
Determine whether the given triangles are similar. If they are, write a similarity statement and name the criterion you used. If the triangles are not similar, explain why.
△JKL△JKL and △PQR△PQR are not similar because the corresponding sides of the triangles are not in proportion.