Sat math Essentials

I.
a
< 
b
II.
c
135°
III.
b
< 
c
a.
I only
b.
III only
c.
I and III only
d.
II and III only
e.
I, II, and III 
Answer
c.
To solve, you must determine the value of the third angle of the triangle and the values of
a
,
b
, and 
c.
The third angle of the triangle 
180°
95°
50°
35° (because the sum of the measures of the
angles of a triangle are 180°).
a
180 
95 
85 (because 
∠
a
and the angle that measures 95° are supplementary).
b
180 
50 
130 (because 
∠
b
and the angle that measures 50° are supplementary).
c
180 
35 
145 (because 
∠
c
and the angle that measures 35° are supplementary).
Now we can evaluate the three statements:
I:
a
< 
b
is TRUE because 
a
85 and 
b
130.
II:
c
135° is FALSE because 
c
145°.
III:
b
< 
c
is TRUE because 
b
130 and 
c
145.
Therefore, only I and III are true.
95°
a°
50°
b°
c°
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Types of Triangles
You can classify triangles into three categories based on the number of equal sides.
■
Scalene Triangle:
no equal sides
■
Isosceles Triangle:
two equal sides
■
Equilateral Triangle:
all equal sides 
You also can classify triangles into three categories based on the measure of the greatest angle:
■
Acute Triangle:
greatest angle is acute
50°
60°
70°
Acute
Equilateral
I
s
o
s
cele
s
S
calene
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■
Right Triangle:
greatest angle is 90°
■
Obtuse Triangle:
greatest angle is obtuse
Angle-Side Relationships
Understanding the angle-side relationships in isosceles, equilateral, and right triangles is essential in solving ques-
tions on the SAT.
■
In 
isosceles triangles
, equal angles are opposite equal sides.
■
In 
equilateral triangles
, all sides are equal and all angles are 60°.
60º
60º
60º
s
s
s
2
2
m
∠
a
= m
∠
b
130°
Obtu
s
e
Right
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■
In 
right triangles
, the side opposite the right angle is called the hypotenuse.
Practice Question
Which of the following best describes the triangle above?
a.
scalene and obtuse
b.
scalene and acute
c.
isosceles and right
d.
isosceles and obtuse
e.
isosceles and acute
Answer
d.
The triangle has an angle greater than 90°, which makes it 
obtuse
. Also, the triangle has two equal sides,
which makes it 
isosceles
.
P y t h a g o r e a n T h e o r e m
The 
Pythagorean theorem
is an important tool for working with right triangles. It states:
a
2
b
2
c
2
, where 
a
and 
b
represent the lengths of the 
legs
and 
c
represents the length of the 
hypotenuse
of a
right triangle.
Therefore, if you know the lengths of two sides of a right triangle, you can use the Pythagorean theorem to
determine the length of the third side.
40
°
100
°
40
°
6
6
Hypotenu
s
e
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Example
a
2
b
2
c
2
3
2
4
2
c
2
9 
16 
c
2
25 
c
2
25
c
2
5 
c
Example
a
2
b
2
c
2
a
2
6
2
12
2
a
2
36 
144
a
2
36 
36 
144 
36
a
2
108
a
2
108
a
108
a
12
6
4
c
3
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Practice Question
What is the length of the hypotenuse in the triangle above?
a.
11
b.
8
c.
65
d.
11
e.
65
Answer
c.
Use the Pythagorean theorem:
a
2
b
2
c
2
, where 
a
7 and 
b
4.
a
2
b
2
c
2
7
2
4
2
c
2
49 
16 
c
2
65 
c
2
65
c
2
65
c
Pythagorean Triples
A 
Pythagorean triple
is a set of three positive integers that satisfies the Pythagorean theorem,
a
2
b
2
c
2
.
Example
The set 3:4:5 is a Pythagorean triple because:
3
2
4
2
5
2
9 
16 
25
25 
25
Multiples of Pythagorean triples are also Pythagorean triples.
Example
Because set 3:4:5 is a Pythagorean triple, 6:8:10 is also a Pythagorean triple:
6
2
8
2
10
2
36 
64 
100
100 
100
7
4
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Pythagorean triples are important because they help you identify right triangles and identify the lengths of
the sides of right triangles.
Example
What is the measure of
∠
a
in the triangle below?
Because this triangle shows a Pythagorean triple (3:4:5), you know it is a right triangle. Therefore,
∠
a
must
measure 90°.
Example
A right triangle has a leg of 8 and a hypotenuse of 10. What is the length of the other leg?
Because this triangle is a right triangle, you know its measurements obey the Pythagorean theorem. You could
plug 8 and 10 into the formula and solve for the missing leg, but you don’t have to. The triangle shows two parts
of a Pythagorean triple (?:8:10), so you know that the missing leg must complete the triple. Therefore, the sec-
ond leg has a length of 6.
It is useful to memorize a few of the smallest Pythagorean triples:
3:4:5
3
2
+ 4
2
= 5
2
6:8:10
6
2
+ 8
2
= 10
2
5:12:13
5
2
+ 12
2
= 13
2
7:24:25
7
2
+ 24
2
= 25
2
8:15:17
8
2
+ 15
2
= 17
2
8
10
?
3
5
a
4
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Practice Question
What is the length of
c
in the triangle above?
a.
30
b.
40
c.
60
d.
80
e.
100
Answer
d.
You could use the Pythagorean theorem to solve this question, but if you notice that the triangle shows
two parts of a Pythagorean triple, you don’t have to. 60:
c
:100 is a multiple of 6:8:10 (which is a multiple
of 3:4:5). Therefore,
c
must equal 80 because 60:80:100 is the same ratio as 6:8:10.
45-45-90 Right Triangles
An 
isosceles right triangle
is a right triangle with two angles each measuring 45°.
Special rules apply to isosceles right triangles:
■
the length of the hypotenuse 
2
the length of a leg of the triangle
45°
45°
x
x
x
2
45°
45°
60
100
c
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■
the length of each leg is 
the length of the hypotenuse
You can use these special rules to solve problems involving isosceles right triangles.
Example
In the isosceles right triangle below, what is the length of a leg,
x?
x
the length of the hypotenuse
x
28
x
x
14
2
28
2
2
2
2
2
2
28
x
x
45°
45°
c
c
2
2
c
2
2
2
2
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Practice Question
What is the length of
a
in the triangle above?
a.
b.
c.
15
2
d.
30
e.
30
2
Answer
c.
In an isosceles right triangle, the length of the hypotenuse 
2
the length of a leg of the triangle.
According to the figure, one leg 
15. Therefore, the hypotenuse is 15
2
.
30-60-90 Triangles 
Special rules apply to right triangles with one angle measuring 30° and another angle measuring 60°.
■
the hypotenuse 
2 
the length of the leg opposite the 30° angle
■
the leg opposite the 30° angle 
1
2
the length of the hypotenuse
■
the leg opposite the 60° angle 
3
the length of the other leg
You can use these rules to solve problems involving 30-60-90 triangles.
60
°
30
°
2
s
s
3 
s
15
2
2
15
2
4
45
°
15
15
45
°
a
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Example
What are the lengths of
x
and 
y
in the triangle below?
The hypotenuse 
2 
the length of the leg opposite the 30° angle. Therefore, you can write an equation:
y
2 
12
y
24
The leg opposite the 60° angle 
3
the length of the other leg. Therefore, you can write an equation:
x
12
3
Practice Question
What is the length of
y
in the triangle above?
a.
11
b.
11
2
c.
11
3
d.
22
2