ll-6 The Pythagorean Theorem Objective To use the Pythagorean theorem and its converse to solve geometric problems. The Pythagorean theorem can be used to find the lengths of sides of right triangles. The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side. The other two sides of a right triangle are called the legs of the triangle. The Pythagorean Theorem In any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs. For the triangle showno a2+f=c2. The diagrams below suggest a proof of the Pythagorean theorem. Each diagram shows a square, (a + b) units on a side, divided into other figures. The diagrams suggest different expressions for the area of tjre square. Equating these expressions leads to the equation az + b2 : c2. I ( a2 a III IVC b2 I (u i b)1: az + b2 + +(!"a) (ctt hf :c2+o(tr*) az + b2 + 4(+"b): c2 + o(+*).'.az+b2:c2 EXample I ttre length of one side of a right triangle is 28 cm. The length of the hypotenuse is 53 cm. Write and solve an equation for the length of the unknown side. Solution cf + b2: c2 o2-12-62 o: {r:tt :f*-e - 1/2f09-7u: \,M : 45 (check on next page.) Rational untl Irrational Numbers 529
Check: 282 + 452 a 532 2809:2809 "/.'. the length of the third side of the right triangle is 45 cm. Answer To draw a line segment with a length ot.v1 units, draw aright triangle whose legs are each I unit long, as shown in the following diagram. Then: a2+b2:c2 12 + 12: c2 l*l:c2 2:c2 -r\/1: c.'. the length of the hypotenuse is V2 units. I The following diagrams show that a segment V2 units long can be used to construct a segment V3 units long, that a segment V3 units long can be used to construct a segment V4 units long, and so on. A series of such triangles can be used to locate inational square roots such as \/r, f1, and V5 on the number llne. The arcs are drawn to transfer the length of the hypotenuse of each triangle to the r-axis. Note that -V) is tocated \/i units to the left of O. The converse of the Pythagorean theorem is also true. It can be used to see if a given triangle is a right triangle. Converse of the Pythagorean Theorem If the sum of the squates of the lengths of the two shorter sides of a triangle is equal to the square of the length of the longest, then the triangle is a right triangle. The right angle is opposite the longest side. 530 Chapter I I
Example 2 State whether or not the three given numbers could represent the lengths of the sides of a right triangle. a. 8, 15, l7 b. 16, 24,30 Solution a. a2 + b2 : cz g2+1522172 64+225 4289 289:289 \.'. 8, 15, and l7 could form a right triangle. Answer b. a2+h2:c2 162+242!302 256+5761900 832 + 9W.'. 16, 24, and 30 could not form a right triangle. Answer Example 3 To the nearest hundredth, what is the length of a diagonal of a rectangle whose width is 18 cm and whose length is 30 cm? Solution a2+b2:c2 TTE:, "v/tdr+tdt: c t/tzsq :, f7r? -z+ :, 6\/34: c 6(5.831) - 6 34.99 - c l8 cm Check: lg2 + 302 L e+.g9)2 1224: 1224.3.'. the length of a diagonal of the rectangle is 34.99 cm. Answer Oral Exercises Evaluate. L feie 2. f* T 3. \/T? -= State and solve an equation for the length of the unknown side. 4. a2 + 6z :c2 (? )2+(? )2:c2,> -^2,, -^ a2+ (?)2+. -( bz :c2 6z :(?)2 b2: 2 b:? Rational and lrrationql Numbers 53f
ftitten Exercises In Exercises 1-10, refer to the triangle at the right. Find the missing length correct to the nearest hundredth. A calculator may be helpful. A l. a: lo, b:24, c:? 3. a:8, b: 5, c:? 5. a:8, b:8, c :? 7. a: "!,b:21,c:29 g. a:6,b: 1,c:40 2. a:5,b:12,c:? 4. a: 13, b:9, c : 2 6, a:16,b:8,c: '! 8. a: J-, b: ll, c: 17 lo.a:5,b:?,c:8 State whether or not the three given numbers could represent the lengths of the sides of a right triangle. ll. 20,21,29 12. 3,9, rl 13. 12, 16,20 14. 16,32,36 15. 15, 20,25 16. 17, 34,39 B 17. 2a,3a,4a 19. 8a, l5a, l1a tr 18. 3a,4a,5a 2O. 6a,7a,8a In Exercises 2l-26, refer to the diagram for Exercises 1-10. Find the missing length correct to the nearest hundredth. / 2l. a: b: 12, 23. a: 18, b:!o,, :, 2s. a: tu, u:20, c:? 22. a:75,b:to,r: 24. a: iu, u: 14, c: 26. a: tru, u:28, c :??? In Exercises 27-30, refer to the diagram for Exercises l-10. Find a and D correct to the nearest hundredth. C 27. a: b, c:60 29. a:!u, c : zo 28. a:3b, c:20 30. a:!u,, : sz Gomputer Exereises Write a BASIC program that will report whether three positive numbers entered with INPUT statements could represent the lengths of the sides of a right triangle. RUN the program for the following series of numbers. 1. 14, 48, 50 532 Chapter 11 2. 0.8, 1.5, 1.7 3. 27,36,45
Problems Make a sketch for each problem. Approximate each square root to the nearest hundredth. A calculator may be helpful. A l. Find the length of each diagonal of a \ i rectangle whose dimensions are 33 cm \ \ by 56 cm. 2. A guywire 20 m long is attached to the top of a telephone pole. The guywire is just able to reach a point on the ground 12 m from the base of the telephone pole. Find the height of the telephone pole. 3. A baseball diamond is a square 90 ft on a side. What is the length from first base to third base? 4. The dimensions of a rectangular doorway are 200 cm by 90 cm. Can a table top with a diameter of 210 cm be carried through the doorway? 5. The base of an isosceles triangle is 18 cm long. The equal sides are each 24 cm long. Find the altitude. B 6. A right triangle has sides whose lengths in feet are consecutive even integers. Determine the length of each side. 7. The longer leg of a right triangle is 6 cm longer than 6 times the shorter leg and also I cm shorter than the hypotenuse. Find the perimeter of the triangle. 8. Find the area of a triangle with three sides of length 4 cm. (Hint: Find the height first.) C 9. What is the length of each diagonal of a cube that is 45 cm on each side? 10. Show that a triangle with sides of lengths x2 + y2,2xy, and x2 - y2 is a right triangle. Assume that -r > y. ll. 12. What is the length of each diagonal of a rectangular box with length 55 cm, width 48 cm, and height 70 cm? Would a meter stick fit in the box? Gary is standing on a dock 2.0 m above the water. He is pulling in a boal that is attached to the end of a5.2 m rope. If he pulls in 2.3 m of rope, how far did he move the boat? Rational and lrrational Numbers 533
tixed Beuiew Enercises Simplify. t. t/@t 2. \/V - roc + 2s s. t/ffi@t$ Write as a fraction in simplest form. 4. (3. to-\2 z a-l 6'9-3-h '0.(-f)' 5. 8. 11. (2x-zr-s1z 5v+ 3-l ' y-j 2r3-lor2-28r 2r-14 6. 9. 12. 78 E-7 { + zarx 2s2+9s-5 5*s Self-Iest 2 Vocahulary irrational numbers (p. 521) Pythagorean theorem (p. 529) Approximate each square root to fhe nearest tenth. Use a calculator or the table at the back of the book ".,rlur.""y. 1. Vbkl 2. \/t7w 3. -Vo-88 Obj. 11-4, p. 521 Simplify. 4. \/14447 s. -Vsiry 6. \/0.2s7 Obj. 1l-5, p.525 Solve. 7. w2:64 8. n2-49:o 9. 36y2-25 :0 10. 11. Find c correct to the nearest hundredth if a: 14 and b : 17. Is a triangle with sides 9, 12, and 14 units long a right triangle? Obj. 11-6, p.529 7 Check your answers with those at the back of the book. Challenge The following "Problem of the Hundred Fowl" dates to sixth-century China: If a rooster is worth 5 yuan, a hen is worth 3 yuan, and 3 chicks are worth I yuan, how many of each, 100 in all, would be worth 100 yuan? Assume that at least 5 roosters are required. 534 Chapter I I
Extra / The Distance Formula The distance between two points on the -r-axis or on a line parallel to that axis is the absolute value of the difference between their x-coordinates. Using the notation A'B' to denote the distance from A' to B', you can write the following: A'B' -- 13-7l: 17-3l: 4 AB:13-71:17-31:4 The distance between two points on the y-axis or on a line parallel to that axis is the absolute value of the difference between their y-coordinates. A'C':12-81 :18-21 :6 AC: 12-8l : 18-2l :6 To find the distance between two points not on an axis or a line parallel to an axis, use the Pythagorean theorem: AC:f(ABf+Grc)J :fo-t+g-4p : v@12 -\56+4 :fn :z\tto -;;4, I o lb(e.4) This method for finding the distance between any two points can be generalized in the distance formula. The Distanca Formula For any points P1(x1, /r) and Pz(xz, yz). Pf z: x/(xz - xllz -r (yz - yt)z P,(x,, ),) Pr(x,12) Rational and lruational Numbers 5J5
Example Solution I Find the distance between points P(-5, 4) and QQ, PQ: (3 - :\/P-+(: =f64+36 : vtoo : l0 Answer Solution2 qr:@ : \4=f + (6f :{64+36 : Vloo : l0 Answer Exercrbes (-5)) +(-2-4) Use the distance formula to find the distance between the given points to the nearest tenth. 1. (-6, 0), (4, 0) 3. (3,4), (e, t) s. (-5, -3), (-e, -6) 7. (-4,6), (s,2) e. (4, -4), (e, -8) tt. (-2, l), (-8, -s) l P({;4) 2. (0, -9), (0, 7) ' 4. (lo, 3), (-4, g) 6. (1, -2), (4, -6) 8. (-4, 7), (-e, -s) 10. (3, -1), (12, -8) 12. (1O, -11), (-9, 3) 13. Use the distance formula to show that the point M(-2, -3) is equidistant from points A(3,9) and B(-7, - l5). 14. Show that the points R(-4, -l), S(3, 6), and T(2,7) are the verrices of a right triangle. Challenge s36 l. On the same set of axes, graph the following line segments to draw a picture. -r=0,2=y=.7 x:18,4=y-9 x*3y:21,o<x36 x-3y:6,6<x<18 x-3y:-27,3<x<15 5x+3y:ll7,15=x=18 x:6,0=y=5 x*3y:6,0=x<6 x-y:-7,0=x=3 x-3y:-9,6=r<18 5x*3Y:45,3<x=6 2. Draw a picture on graph paper using line segments. Write a set of equations and inequalities to describe your picture as in Exercise l. Chapter I I