Unit 6 - Right Triangles
Radical sign
aa
Radicand
Index
b
Simplifying Radicals
04/20/2023Algebra Review
4
9
16
25
36
49
64
Perfect Squares
Simplifying Radicals
81
100
121
144
169
196
225
9 2
=
10 2 =
11 2 =
12 2 =
13 2 =
14 2 =
15 2 =
2 2
=
3 2 =
4 2 =
5 2 =
6 2 =
7 2 =
8 2 =
04/20/2023Algebra Review
4 = 2
Their square roots
81 = 9
100 = 10
121 = 11
169 = 13
144 = 12
196 = 14
225 = 15
16 = 4
25 = 5
36 = 6
64 = 8
49 = 7
9 = 3
Simplifying Radicals
04/20/2023Algebra Review
12
Example 11. Find the largest perfect square that will into the radicand evenly.
2. Write the radicand in factored form using the perfect square as a factor.
3. Simplify the perfect square. (Remove it from the radical.)
Simplifying Radicals
04/20/2023Algebra Review
48
Example 2
2. Write the radicand in factored form using the perfect square as a factor.
3. Simplify the perfect square. (Remove it from the radical.)
Simplifying Radicals
1. Find the largest perfect square that will into the radicand evenly.
04/20/2023Algebra Review
54 311
73 40 512
80
Example 3
Example 4 363
Example 5 63
Example 6
Example 7160
0720
Simplifying Radicals
04/20/2023Algebra Review
5514
106 159
400
Example 8
Example 9 10780
Example 10
360
Example 11
Example 12
3185
1215
20
657
Simplifying Radicals
04/20/2023Algebra Review
1. No perfect square radicand2. No perfect square factor of the radicand3. No fraction as a radicand4. No radical in the denominator
5. No unreduced fractions
Rules for Simplifying Radicals
04/20/2023Algebra Review
Geometric Mean
x is said to be the geometric mean between a and b if and only if :
a x x b
=
Right Triangles 04/20/2023
Example 1
Find the geometric mean between 4 and 25.
x x
=25
4
X 2 = 100
X = 100 = 10
10 is the geometric mean between 4 & 25.
Right Triangles 04/20/2023
Example 2Find the geometric mean between 3 and 6.
x
x=
63
X 2 = 18
X = 18 = 9 · 2 = 3 2
Right Triangles 04/20/2023
Right Triangles 04/20/2023
1. Cut an index card along one of its diagonals.
2. On one of the right triangles, draw an altitude from the right angle to the hypotenuse.
3. Cut along the altitude to form two smaller right triangles.You should now have three right triangles.
Compare the triangles. What special property do they share? Explain.
part 2part 2part 1part 1
If the altitude is drawn to the hypotenuse
in a right triangle, then the length of the
altitude is the geometric mean between
the lengths of the parts of the hypotenuse.
altaltalt
altpart 1=
part 2
Right Triangles 04/20/2023
part 2part 2part 1part 1
If the altitude is drawn to the hypotenuse in a right triangle, then the length of a leg is the geometric mean between the length of the part of the hypotenuse adjacent to the leg and the length of the hypotenuse.
Leg 1part 1
=hyp
Leg 2
hypotenuse
Leg 1
Leg 1
part 2=
hypLeg 2
Leg 2
altalt
Right Triangles 04/20/2023
4 3
x zzy
Example 3
Find the value of
x, y, and z.
Right Triangles 04/20/2023
c 2 = 100
The Pythagorean Theorem
The square of the length of the hypotenuse in a right triangle is equal to the sum of the squares of the lengths of the legs.
c 2 = a 2 + b 2
c 2 = 6 2 + 8 2
c = 100 = 10
c
b
a
8
6c 2 = 36 + 64
EX. 4
Right Triangles 04/20/2023
- 36 - 36
c 2 = a 2 + b 2
12 2 = a 2 + 6 2
108 = a 2
c
b
a12
6144 = a 2 + 36
EX. 5
a = 108 = 36·3 = 6 3
Right Triangles 04/20/2023