𝑅𝑃𝐴𝑅

=𝐴𝑅𝑅𝐸

𝑅𝑃𝐴𝑅

=𝐴𝑅𝑅𝐸

Similarities on right triangle

33

AR

b. If AP=8 and RP=6, find ER and AR

𝑅𝑃𝐴𝑃

=𝐴𝑃𝑃𝐸

68=

8𝑃 𝐸

PE

𝑅𝑃𝐴𝑅

=𝐴𝑅𝑅𝐸

6𝐴𝑅

=𝐴𝑅𝑅𝐸

RE=RP+PE

6𝐴𝑅

=𝐴𝑅100 /6( 𝐴𝑅 )2=100

𝐴𝑅=10

PE

𝐸𝑅=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 – 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 𝑎𝑏)

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