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9.7 Special Right Triangles
There are two SPECIAL Right Triangles that you need to memorize:
 45-45-90 Right Triangle
 30-60-90 Right Triangle
It should be easy to remember the special right triangles when you realize that they are each
related to a familiar REGULAR polygon, with one line of symmetry drawn!
The 45 – 45 – 90 is related to:
The SQUARE!
x
The 30 – 60 – 90 is related to:
The EQUILATERAL TRIANGLE!
30⁰ 30⁰
x
c
x
c
c
b
60⁰
x
A Square with
one DIAGONAL
60⁰
a
a
An Equilateral Triangle
with one ALTITUDE
Special Side Relationship:
Since a = b,
we let both legs = x
Special side Relationship:
Notice that c = 2a,
so if we let a = x, then c = 2x
Remember: (You just can’t afford to forget anything you’ve learned from previous chapters!)
Ch 5.5. p 242: The diagonals of a square bisect the angles at each vertex!
Ch 3.3, p 132: An altitude of the isosceles (and equilateral) triangle bisects the angle from
the vertex, is perpendicular, and is also a median to the side to which it is drawn!
Now that we’ve associated each special right triangle with its regular polygon of origin, we’ll place our focus on
ONE each of the two congruent triangles we created above and see if we can discover a novel way to
approach problems based on our understanding of side relationships possessed by these special right triangles!
Otherwise, it’s the Pythagorean Thm all the time and all the way…
The 45 – 45 – 90
The 30 – 60 – 90
45⁰
?
x
30⁰
2x
?
(isosceles!)
(scalene)
45⁰
x
60⁰
x
Theorem 73: In a 45-45-90 Right Triangle, the lengths of the sides opposite these angles
can be represented by
, respectively.
This means that if a triangle has angles of 45 - 45 - 90 ,
then the two legs are congruent and the hypotenuse is
Proof by the Pythagorean Theorem
equal to the product of a leg length times the square root of 2.
x
c
x
Remember: leg a = leg b, so we use “x” for both legs!
Reality Check!
No matter what the length of the legs is,
the hypotenuse will always be
times greater than that measure
if the triangle has angles that measure 45 – 45 – 90 !
Side Ratio for 45 – 45 – 90
In other words:
In a 45 – 45 – 90 triangle:
 The sides opposite the 45⁰ angles are equal. (Ch 3, Thm 21!
If two angles of a triangle are
congruent, then the sides opposite those angles are also congruent!
 The side opposite the 90⁰ angle is the longest side and hypotenuse!
(It makes
sense that the side opposite the largest angle would have to be the longest, … right?
Official Rules for the 45 – 45 – 90 Right Triangle:
 If you know the leg length, multiply by
to get the hypotenuse length.
 If you know the hypotenuse length, divide it by
to find the leg length.
45 – 45 – 90 Right Triangle Examples :
Problem 1: a = 9, b = 9, c = ?
a, b, c
x, x,
9, 9,
x
9
c
9
Hmmm… looks like x = 9!
x
(We know the legs, need hypotenuse!)
Solution: Multiply a leg length, 9, by
to find the hypotenuse:
Problem 2: a = ?, b = ?, c = 8
a, b, c
x, x,
x, x, 8
x
8
Hmmm… looks like
x
(We know the hypotenuse, need legs!)
Solution: Divide the hypotenuse, 8, by
Problem 3:
a, b, c
x, x, c
,
, c
to get leg a = 4 and leg b = 4
c
x
Hmmm… looks like c =
that’s
x
(We have legs, need hypotenuse!)
Solution: Multiply a leg length,
Problem 4:
a, b, c
x, x,
x, x,
, by
to get hypotenuse 14.
x
x
Hmmm… looks like x = 5!
(We have hypotenuse, need legs!)
Solution: Divide
by
to get legs of length 5.
Theorem 72: In a 30 – 60 – 90 Right Triangle, the lengths of the sides opposite these angles
can be represented by
, respectively.
This means that if a triangle has angles of 30⁰ - 60⁰ - 90⁰,
then the hypotenuse is twice the length of the short leg
We let the short leg = x and hypotenuse = 2x!
So are you wondering where we get
for the long leg?
Proof by the Pythagorean Theorem
a = x and c = 2x
30⁰
b
2x
long leg!
!
60⁰
x
short leg!
Do you remember why c = 2a?
hint, hint…
Reality Check!
No matter what the length of the short leg is,
the hypotenuse will always be twice that measure, and the long leg
times longer if the triangle has angles that measure 30 – 60 – 90 !
Side Ratio for 30 – 60 – 90
Official Rules for the 30 – 60 – 90 Right Triangle:

If you know the short leg length, double it to get the hypotenuse length.

If you know the hypotenuse length, divide it by 2 for the short leg length.

If you know the short leg length, multiply by

If you know the hypotenuse, divide by 2 to get the short leg, then multiply by
leg.
to find the long leg length.
for the long
In other words…
In a 30-60-90 triangle,
 The side opposite the 30 angle is the shortest leg and is equal to
the
hypotenuse.
 The side opposite the 60 angle is the longer leg and is equal to
the hypotenuse times
30 – 60 – 90 Right Triangle Examples:
Problem 1:
a = 4, b = ?, c = ?
30⁰
a,
x,
4,
4,
b,
c
, 2x
, 2x
,8
b
c
2x
2(4) = 8
4(
)
60⁰
4
x
Hmmm… x = 4, so this one is super EASY!
x=4
Solution: We have the short leg, 4, so double for hypotenuse = 8,
and multiply by
for the long leg =
!
Problem 2: a = ?, b = ?, c = 10
a,
x,
x,
5,
b,
(Pssst…start at c and follow the arrows!
This is going to be so easy!)
c
2x
10
10
30⁰
10
√3 times 5
60⁰
x
5
Solution: We have the hypotenuse (c), so divide by 2 for short leg length (a),
then multiply that by
for long leg length (b)!
30 – 60 – 90 Check for Comprehension!
To get back to the short leg (opposite the 30⁰) from the long leg (opposite the 60⁰)
divide by ______.
Once you know the short leg, multiply it by ______ to get the hypotenuse length.
Problem 3: Leg opposite the 60⁰ angle (long leg) =
Mathematically
a,
Geometrically
b,
c
x,
2x
x,
2x
30⁰
2x
,
c = 2x
a = x = coefficient of
a = x = the
coefficient of
c = 2x
60⁰
x
Remember! You can ALWAYS check your results by substituting them back into the
Pythagorean Theorem!
The last one might make you a little uncomfortable with all the fractions involved, but if you really
think about what you are doing, it isn’t that hard to check after all ….
Problem #3 CHECK:
a=
b=
c=
Did you notice that the measures all have a denominator of 3?
Easy breezy! Just multiply each term by three and be rid of the fractions!
Side Ratios the same?
(1x :
x
Now we have: a = 14, b =
Let’s use this:
Does 2
Let’s crunch the numbers:
:
2x)
Yep!
, and c = 28
(it will be quicker as we plug in a, b, and c!)
?
=
Answers for 30-60-90 Check for Comprehension: 1.
Yep, bingo!
2. 2