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Simplifying Radicals: a) 12 b) 60 c) 112 d) � �232 e) � �227

Solve: a) 942 �x b) 8016 2 � a c) 2169 c �

9.1 PYTHAGOREAN THEOREM (right triangles) If a triangle is a right triangle then 222 cba �

a, b are the legs c * c is called the hypotenuse (side opposite the right angle) a * c is also the longest side b 222 cba � c 5 222 75 c � 7

1. h = _________ 2. r = ________

3. Find the area of the triangle. Area = _________ 4.What is the length of the diagonal shown in the rectangular box below? Length = _________

3

9.2 IS THE CONVERSE TRUE? Yes, If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle is a right triangle. If 222 cba � , then the triangle is a right triangle. The three sides that work in the Pythagorean formula are called Pythagorean Triples. Examples of Pythagorean triples: (remember longest side is the hypotenuse) 3-4-5 9-12-15 12-16-20

222 cba � 222 cba � 222 cba �

2. Is 'PYT a right triangle? (yes/no) = _________

Double, Triple,.... The following are examples of Pythagorean Primitives (Three lengths of a right triangle with no common factors) :

3-4-5 8-15-17 5-12-13 7-24-25 Create two new right triangle side lengths for each of the primitives.

3-4-5 8-15-17 5-12-13 7-24-25

Double

Triple

12. Find the area of a right triangle with a hypotenuse that measures 17 cm and one leg that measures 15 cm. 13. How high up on a build ing will a 15-foot ladder reach if the foot of the ladder is placed five feet from the build ing?

5

21

X

X X

Equilateral

30-60-90

X

X

9.3 TWO SPECIAL RIGHT TRIANGLES Number one: Isosceles Right Triangle

ca

ca

caa

�

2

2 22

222

a a

45 D 45 D

c

1. c = _________ 2. q = _________

Number two: 30-60-90 triangle 60 Label the legs as : Shortest, Middle, Longest 30 Fill in the Angles:

Lets rewrite it w ithout fractions: 2y 60 y 1. a= 2.Find the area of the triangle below. Area = _________

10 60

a

2. Find b using the quadradic formula

222 cba � 30 b=_______ 2y 60 y 3. r = _________

7

4. What is the length of the hypotenuse of a 30-60 right triangle with a longer leg of length 16 m?

5. Find the area of an isosceles triangle with a base length of 12 cm and each of the congruent sides having lengths 10 cm Area = _________

6. Find the area of an equilateral triangle with sides measuring 6 meters

5cm

x yt

5cm

5cm

10cm 3

2of 35

31

of 35

x yt

5cm

35

Isosceles Right Triangle 30-60-90 triangle 30 a a 3x 2x

45 D 45 D 60 2a x In an Isosceles triangle, the median from the vertex angle to the base is also a perpendicular bisector, angle bisector, and an altitude. The base angles are also congruent. Vertex Angle

Leg Leg

Base Angle Base Angle Base 1. The following is an equilateral triangle. find x= _______________y=______________ Look at the centroid packet!

9

6cm

2. Find the area of the equilateral triangle using 30-60-90 triangles. 3. AB = 10 cm, Find the area of the equilateral triangle( altitude=median=angle bisector). Find the area of the circumscribed circle. (Hint: the perpendicular bisector of any cord goes through the center of the circle). A B C 4. AB = 5 3 cm, Find the area of the equilateral triangle, the area of the Circumscribed circle, and the area of the inscribed circle.

9.4 WORD PROBLEMS A 25-foot ladder is placed against a build ing. The bottom of the ladder is 7 feet from the build ing. If the top to the ladder slips down 4 feet, how many feet will the bottom slide out? No, it is not 4 fet. This is a two-step problem, so draw it with two right triangles.

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D

9.5 DISTANCE IN COORDINATE GEOMETRY Find the distance between the points (3 , 1) and (6 , 5) using a right triangle. Hint: 1. plot the points, 2. use the line between the two points as the hypotenuse.

Now we are goint to find the formula

222 cba �

Distance formula: D =

1. Find the distance between (1, 0) and (3, 2). Leave your answer in simple radical form.

2. Find the distance between (-1, 2) and (-3, 4). Leave your answer in simple radical form.

D

D

3.Find the distance between (20, 5) and (4, -3). Leave your answer in simple radical form.

4. Find the distance between (4, -3) and (-3, -2). Leave your answer in simple radical form.

5. Find the perimeter of triangle CDF with vertices C( 2,4) , D ( 8,12), and C(24,0). 6. If the d istance from point (x , 7) to (3 ,11) is 5, then find x.

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� �2,1

� �yx,

� � � �� � � � 222

222

222

321

rkyhx

yx

cba

���

���

�

3

EQUATION OF A CIRCLE: (x,y) represents any point on the circle. What we are looking for is an equation for the circle. If the d istance between the points (x , y) and ( 1, 2 ) is 3, then find x and y.

1. first mark the point (1 , 2) and find the points that are 3 units away. What shape does it make?

2. Plug the given information into the Pythagorean theorem. This is the equation that answers the question.

Equation for the given info

Equation of a circle where:

� �kh, =________________ r =__________________

1. Find the equation of the circle with : a) center at (-5,4) and a rad ius of 3 b) center at (0,0) and a rad ius of 10 2. A circle with center (–3, 5) passes through (–9, –3). Find the circumference. Leave your answer in terms of S.

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9.6 CIRCLES AND THE PYTHAGOREAN THEOREM Two previous conjectors that create right triangles: 1. A tangent to a circle is perpendicular to 2. Angles inscribed in a semicircle are right angles. the rad ius drawn to the point of tangency. (the two tangents are also congruent)

2. Find the shaded area. AB = 6 3 cm

3. Find the shaded area. Area = _________