KA Geometry Unit 4
Unit 4: Similarity
Definitions of similarity:
In order to solve missing values of fractions, we need to do the below:
Take this equation for example:
8/10 = 6/m
We must multiply the sum of both denominators on both sides, so:
8/10 10m = 6/m 10m
8m = 60
M = 7.5
Similarity: Two figures are similar if you can map one figure onto another through a series of rigid transformations and/or dilations.
This means their shapes are the same, but their sizes may differ.
Furthermore, if a figure is congruent, it will ALWAYS be similar.
Rational Equations
Take the equation:
x+19-x=23
To get X by itself, we multiply both sides by 9-x. We get:
X + 1 = -23x + 6
53x = 5
5x = 15
X = 3
Introduction to triangle similarity:
Though two triangles may be different in size, they may be similar. Imagine one triangle is bigger than the other, but to get the size the shape, you only have to multiply it 3x on all sides. This is similarity.
Similar triangles have corresponding angles, scaled up/down versions, ratio between sides is constant.
Triangle Similarity Postulates/Criteria
The postulates used with Similarity are: AA (angle-angle) and SSS (side-side-side), SAS (side, angle side)
Similar triangles have:
Corresponding angles (AA)
Corresponding sides (and equal ratio) (SSS) - ratio between sides is equal, not the actual length.
Two corresponding sides and an corresponding angle in between those two sides (SAS). The ratio between the two sides is the same (NOT LENGTH).
Therefore, if triangles have either AA, SAS, or SSS, they are congruent. We don’t need ASA or AAS because we already know the triangles are congruent based on AA.
Similarity:
If we can map one figure onto the other using a sequence of rigid transformations and dilations, then the figures are similar.
two figures have the same shape but may be different sizes
Solving similar triangles:
If we wanted to find a line segment of a triangle from two triangles, we would first need to deduce that the two triangles are similar.
If the two triangles are similar, then we can move on. Image these two triangles:
TRIANGLE CBD ~ TRIANGLE CAE
We can use the side lengths we know to find out what CE is:
CB = 5, BA = 3, CD = 4,
In order to find the line of the line segment, we do this:
Side of Triangle 1/ Total Length (triangle 2) = Side Triangle 2 / Total Length (triangle 2)
The exact side you are using to calculate doesn’t matter, it just has to be consistent.
⅝ = 4/CE
5CE = 32
CE = 32/5
Solving Similar Triangles: Same Side
Let's say we are trying to solve for BC with these two triangles:
ABC ~ BDC
Using the formula: Side of Triangle 1/ Total Length (triangle 2) = Side Triangle 2 / Total Length (triangle 2)
AC/BC = BC/DC
8/BC = BC/2
16 = (BC)^2
BC = 4
Angle bisector theorem:
The Angle Bisector Theorem states that in a triangle, the angle bisector divides the opposite side into two segments that are proportional to the other two sides of the triangle.
Mathematically, if an angle bisector in triangle ABC intersects side BC at D, then:
(AB / AC) = (BD / DC)
Where:
AB and AC are the two sides of the triangle.
BD and DC are the two segments formed by the bisector on the opposite side.
For example, in this problem, we’d find x via this equation:
12/18 = 6/x
We divide equal corresponding sides, then set them equal to each other. The answer is:
108 = 12x
9 = x
X = 9
When choosing what angles to divide, just make sure you are going the same direction from the bisector. So if you chose to go left for the bottom lengths you must also go right for the side lengths.
Solving problems with similar & congruent triangles:
Solving missing side lengths with similar triangles using the formula for similarity:
Side Length Triangle 1 / Total Length = Side Length Triangle 1 / Triangle 1 Total Length
Proving relationships using similarity:
Proving relationship between triangles using similarity.
Solving modeling problems with similar & congruent triangles:
Golden Ratio = 1.61803….
1 + 1/Golden Ratio = Golden Ratio