9.1 PYTHAGOREAN THEOREM (right triangles)
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1 Simplifying Rdicls: ) 1 b) 60 c) 11 d) 3 e) 7 Solve: ) x 4 9 b) c) PYTHAGOREAN THEOREM (right tringles) c If tringle is right tringle then b, b re the legs * c is clled the hypotenuse (side opposite the right ngle) * c is lso the longest side b b 5 c h =. r = 1
2 3. Find the re of the tringle. Are = 4.Wht is the length of the digonl shown in the rectngulr box below? Length =
3 9. IS THE CONVERSE TRUE? Yes, If the lengths of the three sides of tringle stisfy the Pythgoren eqution, then the tringle is right tringle. If b, then the tringle is right tringle. The three sides tht work in the Pythgoren formul re clled Pythgoren Triples. Exmples of Pythgoren triples: (remember longest side is the hypotenuse) b b b. Is PYT right tringle? (yes/no) = Double, Triple,... The following re exmples of Pythgoren Primitives (Three lengths of right tringle with no common fctors) : Crete two new right tringle side lengths for ech of the primitives. Double Triple
4 1. Find the re of right tringle with hypotenuse tht mesures 17 cm nd one leg tht mesures 15 cm. 13. How high up on building will 15-foot ldder rech if the foot of the ldder is plced five feet from the building?
5 9.3 TWO SPECIAL RIGHT TRIANGLES Number one: Isosceles Right Tringle c 1. c =. q = 1 Number two: tringle 60 Lbel the legs s : Shortest, Middle, Longest 30 Fill in the Angles: X X X X Equilterl X
6 Lets rewrite it without frctions: y y =.Find the re of the tringle below. Are = Find b using the qudrdic formul 30 b b= y y r =
7 4. Wht is the length of the hypotenuse of right tringle with longer leg of length 16 m? 5. Find the re of n isosceles tringle with bse length of 1 cm nd ech of the congruent sides hving lengths 10 cm Are = 6. Find the re of n equilterl tringle with sides mesuring 6 meters 7
8 Isosceles Right Tringle tringle 30 x 3 x x In n Isosceles tringle, the medin from the vertex ngle to the bse is lso perpendiculr bisector, ngle bisector, nd n ltitude. The bse ngles re lso congruent. Vertex Angle Leg Leg Bse Angle Bse Bse Angle 1. The following is n equilterl tringle. find x= y= 10cm 5 3 of cm 1 of y t x Look t the centroid pcket! 5cm 5cm y t x 5cm
9 . Find the re of the equilterl tringle using tringles. 6cm 3. AB = 10 cm, Find the re of the equilterl tringle( ltitude=medin=ngle bisector). Find the re of the circumscribed circle. (Hint: the perpendiculr bisector of ny cord goes through the center of the circle). A B C 4. AB = 5 3 cm, Find the re of the equilterl tringle, the re of the Circumscribed circle, nd the re of the inscribed circle. 9
10 9.4 WORD PROBLEMS A 5-foot ldder is plced ginst building. The bottom of the ldder is 7 feet from the building. If the top to the ldder slips down 4 feet, how mny feet will the bottom slide out? No, it is not 4 fet. This is two-step problem, so drw it with two right tringles.
11 9.5 DISTANCE IN COORDINATE GEOMETRY Find the distnce between the points (3, 1) nd (6, 5) using right tringle. Hint: 1. plot the points,. use the line between the two points s the hypotenuse. Now we re goint to find the formul b D Distnce formul: D = 1. Find the distnce between (1, 0) nd (3, ). Leve your nswer in simple rdicl form. D. Find the distnce between (-1, ) nd (-3, 4). Leve your nswer in simple rdicl form. D 11
12 3.Find the distnce between (0, 5) nd (4, -3). Leve your nswer in simple rdicl form. 4. Find the distnce between (4, -3) nd (-3, -). Leve your nswer in simple rdicl form. 5. Find the perimeter of tringle CDF with vertices C(,4), D ( 8,1), nd C(4,0). 6. If the distnce from point (x, 7) to (3,11) is 5, then find x.
13 EQUATION OF A CIRCLE: (x,y) represents ny point on the circle. Wht we re looking for is n eqution for the circle. If the distnce between the points (x, y) nd ( 1, ) is 3, then find x nd y. 1, 3 x, y 1. first mrk the point (1, ) nd find the points tht re 3 units wy. Wht shpe does it mke?. Plug the given informtion into the Pythgoren theorem. This is the eqution tht nswers the question. b x 1 y 3 Eqution for the given info x h y k r Eqution of circle where: h, k = r = 1. Find the eqution of the circle with : ) center t (-5,4) nd rdius of 3 b) center t (0,0) nd rdius of 10. A circle with center ( 3, 5) psses through ( 9, 3). Find the circumference. Leve your nswer in terms of. 13
14
15 9.6 CIRCLES AND THE PYTHAGOREAN THEOREM Two previous conjectors tht crete right tringles: 1. A tngent to circle is perpendiculr to. Angles inscribed in semicircle re right ngles. the rdius drwn to the point of tngency. (the two tngents re lso congruent). Find the shded re. AB = 6 3 cm 3. Find the shded re. Are = 15
16
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