Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion . In other words, similar triangles are the same shape, but not necessarily the same size.
The triangles are congruent if, in addition to this, their corresponding sides are of equal length.
The side lengths of two similar triangles are proportional. That is, if Δ U V W is similar to Δ X Y Z , then the following equation holds:
U V X Y = U W X Z = V W Y Z
This common ratio is called the scale factor .
The symbol ∼ is used to indicate similarity.
Example:
Δ U V W ∼ Δ X Y Z . If U V = 3 , V W = 4 , U W = 5 and X Y = 12 , find X Z and Y Z .
Draw a figure to help yourself visualize.
Write out the proportion. Make sure you have the corresponding sides right.
3 12 = 5 X Z = 4 Y Z
The scale factor here is 3 12 = 1 4 .
Solving these equations gives X Z = 20 and Y Z = 16 .
The concepts of similarity and scale factor can be extended to other figures besides triangles.
Two triangles are similar if they have:
- all their angles equal
- corresponding sides are in the same ratio
But we don't need to know all three sides and all three angles ...two or three out of the six is usually enough.
There are three ways to find if two triangles are similar: AA, SAS and SSS:
AA
AA stands for "angle, angle" and means that the triangles have two of their angles equal.
If two triangles have two of their angles equal, the triangles are similar.
So AA could also be called AAA (because when two angles are equal, all three angles must be equal).
SAS
SAS stands for "side, angle, side" and means that we have two triangles where:
- the ratio between two sides is the same as the ratio between another two sides
- and we we also know the included angles are equal.
If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.
In this example we can see that:
- one pair of sides is in the ratio of 21 : 14 = 3 : 2
- another pair of sides is in the ratio of 15 : 10 = 3 : 2
- there is a matching angle of 75° in between them
So there is enough information to tell us that the two triangles are similar.
Using Trigonometry
We could also use Trigonometry to calculate the other two sides using the Law of Cosines:
In Triangle ABC:
- a2 = b2 + c2 - 2bc cos A
- a2 = 212 + 152 - 2 × 21 × 15 × Cos75°
- a2 = 441 + 225 - 630 × 0.2588...
- a2 = 666 - 163.055...
- a2 = 502.944...
- So a = √502.94 = 22.426...
In Triangle XYZ:
- x2 = y2 + z2 - 2yz cos X
- x2 = 142 + 102 - 2 × 14 × 10 × Cos75°
- x2 = 196 + 100 - 280 × 0.2588...
- x2 = 296 - 72.469...
- x2 = 223.530...
- So x = √223.530... = 14.950...
Now let us check the ratio of those two sides:
a : x = 22.426... : 14.950... = 3 : 2
the same ratio as before!
Note: we can also use the Law of Sines to show that the other two angles are equal.
SSS
SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio.
If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
In this example, the ratios of sides are:
- a : x = 6 : 7.5 = 12 : 15 = 4 : 5
- b : y = 8 : 10 = 4 : 5
- c : z = 4 : 5
These ratios are all equal, so the two triangles are similar.
Using Trigonometry
Using Trigonometry we can show that the two triangles have equal angles by using the Law of Cosines in each triangle:
In Triangle ABC:
- cos A = (b2 + c2 - a2)/2bc
- cos A = (82 + 42 - 62)/(2× 8 × 4)
- cos A = (64 + 16 - 36)/64
- cos A = 44/64
- cos A = 0.6875
- So Angle A = 46.6°
In Triangle XYZ:
- cos X = (y2 + z2 - x2)/2yz
- cos X = (102 + 52 - 7.52)/(2× 10 × 5)
- cos X = (100 + 25 - 56.25)/100
- cos X = 68.75/100
- cos X = 0.6875
- So Angle X = 46.6°
So angles A and X are equal!
Similarly we can show that angles B and Y are equal, and angles C and Z are equal.
Copyright © 2017 MathsIsFun.com
learning task 4: add/subtract the following fractions.write your answer to it's simplest form 1). 8/7-5/6 =2).7/6+4/5=3). 10 5/7-5 2/3=4). 8 5/8+5 3/4 … =
Mr.Covid and his co-workers can deliver 231 coconuts in a day How many coconuts can they deliver in 23 daysquestions1.How many coconuts can Mr.Covid a … nd his-workers deliver in a day2.what is asked i. the problem?3.solve the problem
Activity 1 Domain and Range Direction: Determine the domain and range of the given relation and decide whether the relation is function. Write your an … swer on the box given. 3 POINTS PER ITEM. Set of Ordered pairs 1. {(1,2)(2,3) (3,4) (4,5), (5,6)} 2. {(-8, 2) (1/2, 3/5)} 3. {(7,2)(7,-2)(2,7),(-2,7)} 4. {(1.5, 2.0) (2.0, 1.5) (1.5, 3)(3, 1.5)} 5. {(c, 1) (c, 2) (c, 3) (c, 4)} DOMAIN RANGE FUNCTION OR NOT A FUNCTION
Answer the following l if it is divisible by d-28 A 2,5,10 12,962 2.5/5 3.700 4.2515 d∙r 0.9 d.
find the x-intercept of the rational function y=(x+10)/(x-5)
Answer the following if it is divisible by 2,5,10,1.2'462 2.515 3.700 4.2515 pls need answer pls
ײ-8×-9=0 quadratic equationfind the solution
give am example of a pattern that occurs in nature or in the world that does not involve number
define the following terms nullset
what is the value of 67 840 thousand?Who ever answer this ill follow and brainlest it!None-sence answer=ReportWrong Answer=Report
There are three easy ways to prove similarity. These techniques are much like those employed to prove congruence--they are methods to show that all corresponding angles are congruent and all corresponding sides are proportional without actually needing to know the measure of all six parts of each triangle.
AA (Angle-Angle)
If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion. Picture three angles of a triangle floating around. If they are the vertices of a triangle, they don't determine the size of the triangle by themselves, because they can move farther away or closer to each other. But when they move, the triangle they create always retains its shape. Thus, they always form similar triangles. The diagram below makes this much more clear.
Figure %: Three pairs of congruent angles determine similar triangles In the above figure, angles A, B, and C are vertices of a triangle. If one angle moves, the other two must move in accordance to create a triangle. So with any movement, the three angles move in concert to create a new triangle with the same shape. Hence, any triangles with three pairs of congruent angles will be similar. Also, note that if the three vertices are exactly the same distance from each other, then the triangle will be congruent. In other words, congruent triangles are a subset of similar triangles.Another way to prove triangles are similar is by SSS, side-side-side. If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar.
Figure %: If all three pairs of sides of corresponding triangles are in proportion, the triangles are similarSAS (Side-Angle-Side)
If two pairs of corresponding sides are in proportion, and the included angle of each pair is equal, then the two triangles they form are similar. Any time two sides of a triangle and their included angle are fixed, then all three vertices of that triangle are fixed. With all three vertices fixed and two of the pairs of sides proportional, the third pair of sides must also be proportional.
Figure %: Two pairs of proportional sides and a pair of equal included angles determines similar trianglesConclusion
These are the main techniques for proving congruence and similarity. With these tools, we can now do two things.
- Given limited information about two geometric figures, we may be able to prove their congruence or similarity.
- Given that figures are congruent or similar, we can deduce information about their corresponding parts that we didn't previously know.
Did you know you can highlight text to take a note? x