Answers. Investigation 4. ACE Assignment Choices. Applications. The number under the square root sign increases by 1 for every new triangle.

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1 Answers Investigation 4 ACE Assignment Choices Problem 4. Core, Other Connections 6 Problem 4. Core, 4, Other Applications 6 ; Connections 7, 6, 7; Extensions 8 46; unassigned choices from earlier problems Problem 4. Core 0, Other Connections 6 4; Extensions 47 ; unassigned choices from earlier problems Problem 4.4 Core, Other Extensions 8; unassigned choices from earlier problems Adapted For suggestions about adapting Exercise 8 and other ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units 7 8: Moving Straight Ahead; 6: Filling and Wrapping; 8, : Stretching and Shrinking; : Bits and Pieces I Applications. cm. a. The th triangle has leg lengths unit and " units and hypotenuse length " units. The th triangle has leg lengths unit and " units and hypotenuse length "4 units. The 4th triangle has leg lengths unit and "4 units and hypotenuse length " units. b. sq. unit,? units ",? " units,? units, unit "4,? " units. The number under the square root sign increases by for every new triangle. Or, the area of the nth triangle is.? "n c. is the square root of. So, the hypotenuse length of the 4th triangle is units.. "00 00 = "800 < 8.8 in. 4. "44 6 = "8 <. ft. a. Because = 60,000, the distance is "60,000 < 78 m. b.,00-78 < m 6. a. They are congruent. b. 48,48,08. The diagonal divides the corner angles into two equal angles, so the smaller angles must each be half of 08, or 48. Some students may use a protractor or angle ruler. c. The legs of the right triangle each have a length of unit, and + =. So the diagonal which is the hypotenuse of a right triangle has a length of " units. d. The measures of the angles would still be 48,48, and 08. Because + = 0, the length of the diagonal would be "0 units. (Note: Some students may notice that "0 = "? = ", or that this square is larger than the original by a scale factor of ; thus, the diagonal must be times as long, or " units.) 7. a. All triangles are similar to each other. If corresponding angles of a triangle are congruent, then the triangles are similar. b. The other leg must also be units long because triangles are isosceles. Applying the Pythagorean Theorem we have (hypotenuse) = + = 0, so hypotenuse = "0 = " < 7.07 units. So, the perimeter is + + " < 7.07 units. ACE ANSWERS 4 Investigation 4 Using the Pythagorean Theorem

2 m. The first segment along the ground is the leg of an isosceles right triangle. Because the other leg is m long, this leg also has a length of m. The same argument holds for the last segment along the ground. Therefore, the horizontal portion of cable is,000 - (? ) = 70 m long. Each angled part of the cable is the hypotenuse of an isosceles right triangle with legs of length units. Because + = 40, each angled piece has length "40 <. m. The overall length of the cable is thus < 0.4 m.. ft. Because - = 400, the tallest tree that can be braced is "400 ft, or 0 ft tall at the point of attachment. Adding ft gives a total height of ft. (Note: You can point out to students that this is a -4- Pythagorean Triple with a scale factor of.) 0. About 0. ft. The leg along the bottom of the triangle measures 8 ft. The hypotenuse (from Denzel s eyes to the top of the tower) is twice as long, or 6 ft. Because 6-8 = 0,0, the vertical leg measures "0,0 < 00. ft. Adding the distance from the ground to Denzel s eyes, the tower is about 0. ft tall.. a. ABC, ADE, and AFG are triangles. The measure of angle A, which is in all three triangles, is 608. Angles ACB, AED, and AGF all have measure 0 because the segments that form their sides are perpendicular (one side is horizontal and the other is vertical). So, the third angles of the three triangles ABC, ADE, and AFG must all have measure 08. These triangles are all similar because if corresponding angles of a triangle are congruent, then the triangles are similar. BA 4 b. = =. The length of AC is units AC and, because triangle ABC is a triangle, BA is twice the length of the side opposite the 08 angle, which is AC. Therefore, the length of BA is 4 units. The corresponding ratio for the other two triangles must be the same because the triangles are similar. BC " c. = = ". Possible explanation: In AC a triangle, the length of the side opposite the 608 angle is " times the length of the side opposite the 08 angle, which is AC. AC has length units, so BC has length BC " " units. So, = = ".The AC corresponding ratio for the other two triangles must be the same because the triangles are similar. BC " " d. = =. The corresponding AB 4 ratio for the other two triangles must be the same because the triangles are similar. e. 4 units and " units. Possible explanation: Triangle XYZ fits the description given in the problem. YZ The ratio XZ must be equal to " because XYZ is similar to triangle ABC in part (a). Therefore, YZ = XZ? " = ". In all triangles, the ratio of the hypotenuse to the shortest side is :. So XY =? = 4.. About 8. m. All triangles in the diagram are triangles. The hypotenuse of the large triangle is m (twice the shorter leg that is given). The longer leg is 6#, or "08 < 0.. This last leg can be found with the Pythagorean Theorem or by applying a scale factor of 6 to the triangle in Question A of Problem 4.. Connections. # = ; rational 4. #0.4 = 0.7; rational. # <.; irrational 6. #000 <.6; irrational Z Y 0 Á 4 60 X Looking for Pythagoras

3 7. See Figure. The distance between the cars increases by 78. mi each hour. (Note: Students will probably calculate the distance apart by adding the sum of the squares and taking the square root of that sum.) 8. After hr, the northbound car has traveled 80 mi. Use this distance as one leg of a right triangle and the distance apart (00 mi) as the hypotenuse. Using the Pythagorean Theorem, =,600, so the distance the eastbound car has traveled must be ",600 = 60 mi. This distance was traveled in hr, so the eastbound car is traveling at 0 mph. (Note: This is a -4- right triangle with a scale factor of 0.). = 0.4; terminating 0. = 0.7; terminating 8. = ; repeats 6. =.; terminating 0 8. = ; 08 repeats 4. Right triangle. + 7 = ("74). Right triangle. (") + ("7) = + 7 = = Figure 80 mi Hours 60 mi 00 mi Distance Traveled by Northbound Car (mi) a. " <.66 cm b. c. About B 8. a. Two pairs of corresponding angles are equal, so the triangles are similar. b. Because the triangles are similar, the corresponding sides are proportional. The given side length of the smaller triangle is a third of the corresponding side length of the larger triangle, so the other two side lengths of the smaller triangle must also be a third the length of the corresponding sides of the larger triangle. The sides of the larger triangle are 6 units, units, and # or "7 units (or about. units), so the sides of the smaller triangle are units, unit, and " or units (or about "7.7 units). c. The larger triangle s area is times the smaller triangle s area.. Possible answers: or Distance Traveled by Eastbound Car (mi) 0 Distance Between Cars (mi) Á ACE ANSWERS Á Á Á n 60n 0n 78.n Investigation 4 Using the Pythagorean Theorem

4 ,46 0. Possible answer: 0,000. Possible answer:. False. 0.06? 0.06 = True..?. =. 4. False. 0? 0 = 400. a. About 7. units. AC = 6 units, CD = " units, or 8" units, or about. units. So the perimeter is about , or 7. units. b. Because triangle BDC is a triangle, we can use the length of AC to get the length of AB, which is units, and of BC, which is 6" units. So the perimeter of triangle ABC is ", or about 7.7 units. We could have arrived at this answer without any calculation by noticing that the triangles are similar and the scale factor is. Therefore, the perimeter of triangle ABC is twice the perimeter of triangle ACD. c. The area of triangle ABC is 4 times the area of triangle ACD and 7. 6 = 6 and 7 = 4. Because is between 6 and 4, " is between 6 and and. 4 = 76 and = 6. Because 600 is between 76 and 6, "600 is between 4 and. Extensions 8. a. Fraction ,00,000 Decimal b. Each fraction is equivalent to a repeating decimal. The repeating part is a single digit that is equal to the numerator of the fraction. c or ;....; d..... = = + = = = + = 7. = , = , = A fraction with a denominator of is equal to a repeating decimal. For numerators less than, the repeating part has two digits: either a 0 followed by the number in the numerator if that number is less than 0 or the number in the numerator if that number is greater than = , = , = A fraction with a denominator of is equal to a repeating decimal. For numerators less than, the repeating part has three digits: two 0s followed by the number in the numerator if that number is less than 0; one 0 followed by the number in the numerator if that number is greater than 0 and less than 00; or the number in the numerator if that number is greater than or The bottom of the box has sides of length cm and 4 cm. Because + 4 =, the diagonal of the bottom has length " cm, or cm. Using this as a leg of a right triangle with hypotenuse d, d = + = 6, so d = "6 cm = cm. 48. The bottom has sides of length 6 cm and 7 cm. Because = 8, the diagonal of the bottom has length "8 cm. Using this as a leg of the right triangle with hypotenuse d, d = ("8) + (") = 8 + = 6, so d = "6 cm = 4 cm. 4 Looking for Pythagoras

5 4. a. (.4,.4). Draw a vertical segment from B down to the x-axis to create a triangle ABC. 4 A O y 48 As observed in Exercise 7, in triangles, the length of the hypotenuse is " times the length of the leg. So BC = AC = units, which is " approximately.4 units. So, the coordinates of B are (.4,.4). b. 0. a. The half-circle on the leg of length units has area? p?. <. units. The half-circle on the leg of length 4 units has area? p? < 6. units. The half-circle on the hypotenuse has area? p?. <.8 units. b. The sum of the areas of the half-circles on the legs is equal to the area of the half-circle on the hypotenuse: =.8.. a. Each equilateral triangle can be divided into two triangles. The equilateral triangle on the leg of length units is composed of two right triangles, each with a leg of length. units and a hypotenuse of length units. Because -. = 6.7, the longer leg (which is the height of the equilateral triangle) has length "6.7 <.6 units. This equilateral triangle has an area of about??.6 =. sq. units. The equilateral triangle on the leg of length 4 units is composed of two right triangles, each with a leg of length units and a hypotenuse of length 4 units. Because 4 - =, the longer leg has length " <.46 units. This equilateral triangle C B x has an area of about? 4?.46 = 6. units. The equilateral triangle on the hypotenuse is composed of two right triangles, each with a leg of length. units and a hypotenuse of length units. Because -. = 8.7, the longer leg has length "8.7 < 4. units. This equilateral triangle has an area of about?? 4. = 0.8 units. b. The sum of the areas of the equilateral triangles on the legs is equal to the area of the equilateral triangle on the hypotenuse: = a. Each hexagon can be divided into six equilateral triangles, the areas of which were found in ACE Exercise. The hexagon on the leg of length units has an area of about 6?. =.4 units.the hexagon on the leg of length 4 has an area of about 6? 6. = 4.4 units. The hexagon on the hypotenuse has an area of about 6? 0.8 = 64.8 units. b. The sum of the areas of the hexagons on the legs is equal to the area of the hexagon on the hypotenuse: = Possible answers: ", "40, and p. 4. a. 00x =.... x = x = x = or b. 0x = x = x = 7 7 x = c.,000x =.... x = x = x = or 4. a. "00 6 = "64 = 8 ft b. The farmer is saying that the barn is not perpendicular to the ground. c. " 44 = "8 = ft ACE ANSWERS 4 Investigation 4 Using the Pythagorean Theorem

6 d. Possible answer: She could use a -foot pole that would touch the barn 4 ft high and rest on the ground ft from the base of the barn units. Triangle CDB is similar to triangle ABC, because both have angle B and a right angle. Because + = 6, the length of side BC is "6 = units. The leg of length units on the small triangle corresponds with the leg of length units on triangle ABC,so the scale factor from triangle CDB to triangle ABC is, or.6. Multiplying the side lengths of triangle CDB by.6, side AC has length?.6 =. units and side BA has length?.6 =.8 units. The perimeter of triangle ABC is thus = 78 units. (Note: Students may also calculate that triangle CDB has a perimeter of + + = 0 and then apply the scale factor to find that the perimeter of triangle ABC is 0?.6 = 78.) 7. a. Using the Pythagorean Theorem, the length of half of the edge of the base is units, so the edge length of the base is 6 units. Therefore, the base area is 6 units. b. The surface is made up of 4 congruent triangles plus a base. Each triangle has area ( )(6)(4) = units. So the surface area is 6 + 4() = 84 units. c. The height of the pyramid is found from the right triangle with sides units (half of the base edge) and 4 units (the slant height). We need to solve + h = 4. h is "7 units, or about.6 units. d. ( )(6)(.6) <.8 units. 8. a..8 in.. Because the diameter is 4. in., the radius is. in. The height is 6 in., so the volume is p(.) (6) <.8 in. b. 6p in..7 = r + 6, so r = " in., or about.6 in. So the volume is p( ") (6) = 6p in., or about 8.7 in. Possible Answers to Mathematical Reflections. The Pythagorean Theorem is useful for finding the length of one side of a right triangle if you know the lengths of the other two sides. An example of this is finding the distance between two points when the coordinates of the points are known. We connect the points with a line segment and then use the segment as the hypotenuse of a right triangle. We draw the two legs, find their lengths, and then find the sum of the squares of the lengths. The distance between the two points is the square root of this sum. Another example is finding the length of the diagonal d of a rectangle. If the side lengths of the rectangle are a and b, then the Pythagorean Theorem tells us d = a + b, or d = "a b.. In a triangle, the length of the side opposite the 08 angle is half the length of the hypotenuse. The length of the longer leg is " times the length of the leg opposite the 08 angle. (Note: In a triangle, if the leg opposite the 08 angle has length a, then the hypotenuse has length a. So, the longer leg has length "4a a = "a = a".) Answers to Looking Back and Looking Ahead. a.. units b.." units, or "., or approximately.6 units c. 0 units, " units, and " units d. Triangle B: scale factor is (in other words, triangles A and B are congruent); Triangle F: scale factor from F to A is and from A to F is ; Triangle D: scale factor from D to A is and from A to D is ; Triangle G: scale factor from G to A is " and from A to G is " 6 Looking for Pythagoras

Investigation Find the area of the triangle. (See student text.)

Investigation Find the area of the triangle. (See student text.) Selected ACE: Looking For Pythagoras Investigation 1: #20, #32. Investigation 2: #18, #38, #42. Investigation 3: #8, #14, #18. Investigation 4: #12, #15, #23. ACE Problem Investigation 1 20. Find the area