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π ππ΄π
=π΄π π πΈ
π ππ΄π
=π΄π π πΈ
Similarities on right triangle
33
AR
b. If AP=8 and RP=6, find ER and AR
π ππ΄π
=π΄πππΈ
68=
8π πΈ
PE
π ππ΄π
=π΄π π πΈ
6π΄π
=π΄π π πΈ
RE=RP+PE
6π΄π
=π΄π 100 /6( π΄π )2=100
π΄π =10
PE
πΈπ =1006
=16.67
Solve for x and y
y
x2=(8)(5)X=6.3
y2=(13)(8)y=10.2
Solve for x and y52-32=x2
25 β 9 =x2
16 = x2
4 =x
z
Some Theorems on a Right Triangles
The Pythagorean TheoremIn 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
(a+b)2=c2+
a2+2ab+b2=c2+2ab
a2+b2=c2
(a+b)2 c2 4( 12 ππ)
A Pythagorean triple is a group of three wholeNumbers 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==24
Pythagorean Triple
6
6
c = 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?
Some Theorems on a Right Triangles
The Median TheoremThe median to the hypotenuse of a right triangle is one half as long as the hypotenuse
AC
B
D
CD=
Determine the values of x and y.
x
12
Using the Median theorem
x=
x8
15
y+4
x2=82+152
x2=64+225
x2=289
x=17
y+4=
2y+8=17
2y=9
y=4.5
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 times as long as the opposite the 30o angle.
AC
D
B
60o
30o
BC=AB
AC=BC
The 30-60-90 Triangle Theorem
The 30-60-90 Triangle Theorem
AC
B
60o
30o
x2x
x
Find the measure of the missing sides of the triangle.
30o
60o
8
ac
30o
b
a18
30oc
a
12
The Isosceles Right Triangle Theorem
The Isosceles Right Triangle, the hypotenuse is times as long as either of the legs.
AC
c
B
c=x
x
45
45
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