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Families
30ᵒ 60ᵒ, 90ᵒ
45ᵒ 45ᵒ, 90ᵒ
Resource
Geometry/ Sophomores
When working with right triangles, there
are certain set of groups that you must know. There are
three main sets that you must learn: a 3-4-5 triangle, 512-13 triangle and 7-24-25 triangle. The larger of the
number of each set corresponds to the hypotenuse.
Then the other two are part of each set correspond to
the legs of the triangle.
Another important note is that other groups
of sets can be similar to the three basics ones presented
here. For example a 3-4-5 is similar to a 6-8-10 triangle
because of simplifying ratios between the two triangles.
It is important for you to remember these basic families
because if will save you time rather than doing a
Pythagorean theorem to solve for each side if two sides
are known.
Example
This triangle is similar to
which basic family?
A.
B.
C.
Another special triangle is the 30-60-90
degree triangle. Each degree corresponds with its
own leg. For instance the 30 degree angle
corresponds with the leg across from it which is x.
The 60 degree triangle belongs to the leg across
from it, x square root of 3. And finally the 90
degree, right angle side, corresponds to the 2x leg
(hypotenuse).
Placing a value in for x will change the
length of the sides of the triangles. For example if
x=2, the length of the leg across from 30 degrees
is 2, the leg across from 60 degrees is 2 square
root of 3, and the hypotenuse is 4.
Example
If x=3, what are the lengths of a 30ᵒ
60ᵒ, 90ᵒ triangle?
A.
B.
C.
Finally, a different special triangle is
the 45-45-90 degree triangle. Each degree
corresponds with its own leg. For instance the
45 degree angle corresponds with the two leg
across from it which is x. And finally the 90
degree, right angle side, corresponds to the x
square root of 2 leg (hypotenuse).
Placing a value in for x will change
the length of the sides of the triangles. For
example if x=2, the length of the legs across
from the 45 degrees is 2, and the hypotenuse
is 2 square root of 2. Depending of the value
of x, can change the lengths of the sides. The
legs across the 45 degree are congruent
because the legs and angles equal each other.
Example
 If
A.
x=4, what are the side lengths of a 45-4590 degree triangle?
B.
C.

Image of 3,4,5 triangle on slide 1 is from: http://justcolleges.com/tests/triangles-tutorial-for-satgre-test-preparation/

Image of 30-60-90 degree triangle on slide 1 is from:
http://hotmath.com/hotmath_help/spanish/topics/30-60-90-triangles.html

Image of 45-45-90 degree triangle on slide 1 is from:
http://www.sparknotes.com/testprep/books/newsat/chapter20section4.rhtml

Image of 3,4,5 right triangle on slide 2 from:
http://mathyear2013.blogspot.com/2013/08/another-rule-about-pythagorean-triples.html

Information from slides 2, 4, 6 is from:
http://www.sparknotes.com/testprep/books/newsat/chapter20section4.rhtml

Images of 5-12-13 & 7-24-25 triangle is my own creation.

Image of 30-60-90 degree triangle is my own creation.

Image of 45-45-90 degree triangle is my own creation.
Make sure you are simplifying as low as possible
First, choose two legs to work with.
example: 15 & 36
Second, make them into a fraction.
example: 15/36
Third, simplify the fraction.
example:15/36=5/12
=
Remember that the leg across from the 30ᵒ is
x, the leg across from the 60ᵒ is x square root
of 3, and the leg across the 90ᵒ, which is also
the hypotenuse, is 2x

Remember that the leg across from the
45ᵒ is x, the leg across from the other 45ᵒ is
x, and the leg across the 90ᵒ, which is also
the hypotenuse, is x square root of 2.
=