Similarity on a Right Triangle Right Triangle with altitude Right Triangle Theorem 30-60-90 Triangle Theorem Isosceles Right Triangle Theorem

1. 𝑅𝑃
𝐴𝑅
=
𝐴𝑅
𝑅𝐸

2. 𝑅𝑃
𝐴𝑅
=
𝐴𝑅
𝑅𝐸
Similarities on right triangle
𝐴𝑅 2 =33
AR= 33
b. If AP=8 and RP=6, find ER and AR
𝑅𝑃
𝐴𝑃
=
𝐴𝑃
𝑃𝐸
6
8
=
8
𝑃𝐸
PE=
32
3
𝑅𝑃
𝐴𝑅
=
𝐴𝑅
𝑅𝐸
6
𝐴𝑅
=
𝐴𝑅
𝑅𝐸
RE=RP+PE
6
𝐴𝑅
=
𝐴𝑅
100 6
𝐴𝑅 2 = 100
𝐴𝑅 = 10
PE=
8(8)
6
=
64÷2
6÷2
=
𝐸𝑅 =
100
6
= 16.67

3. Solve for x and y
y
x2=(8)(5)
X=6.3
y2=(13)(8)
y=10.2

4. Solve for x and y
52-32=x2
25 – 9 =x2
16 = x2
4 =x
z

5. Some Theorems on a Right Triangles
The Pythagorean Theorem
In a right triangle, the square of the hypotenuse is equal
to the sum of the squares of the legs.
b
a
ab
b
a
ba
c c
c c
Prove using the Area formula:
Area of a largest square=Area of the smaller square + Area of the 4 right ∆′
𝑠.
(a+b)2=c2+4
1
2
𝑎𝑏
a2+2ab+b2=c2+2ab
a2+b2=c2
(a+b)2 c2 4
1
2
𝑎𝑏

6. A Pythagorean triple is a group of three whole
Numbers that satisfies the equation a2+b2=c2
where c is the greatest number.
Pythagorean Triple
3
4
5 32+42=52
9
12
c
7b
25
=15
24=
252-72=
625-49= 576=24

7. Pythagorean Triple
6
6
c =6 2 37
35
a12=
Determine if the given measures are the length of the
Sides of a right triangle.
a. 9,12,15 b. 12,35,36
A 13-ft ladder is leaning against the wall. The base of
the ladder is 5 ft. from the wall. How high up the wall
does the ladder reach?

8. Some Theorems on a Right Triangles
The Median Theorem
The median to the hypotenuse of a right triangle is
one half as long as the hypotenuse
AC
B
D
CD=
1
2
𝐴𝐵

9. Determine the values of x and y.
x
12
Using the Median theorem
x=
1
2
12 = 6
x
8
15
x2=82+152
x2=64+225
x2=289
x=17
y+4=
1
2
17
2y+8=17
2y=9
y=4.5

10. The 30-60-90 Triangle Theorem
In a 30-60-90 triangle, the side opposite the 30o
angle is half as long as the hypotenuse and the
side opposite the 60o angle is 3 times as long as
the opposite the 30o angle.
AC
D
B
60o
30o
BC=
1
2
AB
AC= 3BC

11. The 30-60-90 Triangle Theorem

12. The 30-60-90 Triangle Theorem
AC
B
60o
30o
x
2x
x 3
Find the measure of the missing sides of the triangle.
30o
60o
8
a
c
30o
b
a
18
30o
c
a
12

13. The Isosceles Right Triangle Theorem
The Isosceles Right Triangle, the hypotenuse is 2
times as long as either of the legs.
AC
c
B
c= 2 x
x
x
45
45