Similar Figures and Solids

Questions on Similar Figures & Solids 1) 50 cm B C x A E D 35 cm 55 cm 65 cm Angle BAC = angle EDC as shown. AB = 50 cm. AE = 65 cm. EC = 35 cm. DC = 55 cm. Find the measure of DE hint: mark the angles in the triangles you're comparing. .solution: 50 cm B C x A E D 35 cm 55 cm 65 cm o C o we mark the common angle C we mark the other equal angles o We compare small DEC to big ABC (note the angles at the ends of the sides) DE (arc to o) AB (arc to o) = DC (arc to c) AC (arc to c) x 50 = 55 100 x = 27.5 cm. 2) Area of right triangle GHI is 420 cm². GI = 74 cm. In right triangle JKL, JL = 20 cm. Given that angle I = angle L (as shown), what is the area of triangle JKL to the nearest cm²? GH I JK L 74 cm 20 cm A = 420 cm² A = ? x x A) 31 cm² ......... B) 57 cm²......... C) 76 cm²......... D) 114 cm². solution: A) 31 cm² is true: k = 20/74 = 10/37 so k² = ( 10/37)² --A = ( 10/37)² × 420 = 30.679 Practice Exercises for Jayme3) Triangle MNO has these characteristics: -angle MNO = 35°, ......... -angle NMO = 45° , ......... -NO = 10 m which of these triangles must be congruent to triangle MNO? 45° 100° 35° Q P R A) C) B) D) 100° 35° Q P R 10 m 45° 100° Q P R 10 m 45° 35° Q P R 10 m solution: B) is correct. angle N = 35° = Q, and angle M = 45° = P, so O = R, NO = QR = 10 m. 4) What must be the value of s, t, and u so that the 2 prisms are equivalent? 4 cm 16 cm 9 cm u s t A) s = 3 cm, t = 2 cm, u = 4 cm. ......... C) s = 4.5 cm, t = 2 cm, u = 8 cm. B) s = 8 cm, t = 4 cm, u = 18 cm........... D) s = 8 cm, t = 3 cm, u = 12 cm. solution: B) s = 8 cm, t = 4 cm, u = 18 cm. since s×t×u must equal 9 × 4 × 16 = 576. Practice Exercises for Jayme5) In quadrilateral ABCD, line segments TU and VW intersect at S. D A W C T S B U V Points T, U, V and W are on quadrilateral ABCD. Quadrilaterals AVST and CWSU are congruent. Complete steps 2 and 3 of the proof that ABCD is a parallelogram. Because the opposite sides of a parallelogram are parallel. Step 4 ABCD is a parallelogram AB //DC Because . . . AD //BC Step 3 ø AVS {øCWS Because . . . ø ATS {øCUS . Step 2 Step 1 Quadrilateral AVST { Quadrilateral CWSU Given Solution: Because the opposite sides of a parallelogram are parallel. Step 4 ABCD is a parallelogram Because alternate angles are equal, so lines are parallel. AB //DC AD //BC Step 3 Because corresponding angles in congruent figures are congruent. ø AVS {øCWS ø ATS {øCUS . Step 2 Step 1 Quadrilateral AVST { Quadrilateral CWSU Given 6) A carpenter is going to cut off the top of a right circular cone with height H = 17.6 dm. and base radius r = 4.8 dm. to use the frustrum or base to build a table. The volume of the frustrum is 106.16 dm³. Find h the height of the small cone he cut from the big one. H = 17.6 dm r = 4.8 dm h = ? frustrum vol. = 106.16 dm³ Practice Exercises for JaymeSolution: First, we find the Volume of the whole cone using V = 13 r2h Volume (big cone) : V = 13 (4.8)2(17.6) = 424.64 Now we find the Volume of the small cone by subtraction: Volume (small cone) : 424.64 – 106.16 = 318.48 dm³. The quotient of 318.48 and 424.64 gives us the level 3 ratio of small to big. We find k ³ = 0.75 which means the ratio of h to H is the cube root of 0.75 = 0.9085. This tells us that h = 0.9085 × 17.6 = 16 dm. 7) The Listenup Company makes audio speakers in 2 sizes. The speakers are similar rectangular prisms. The small one is 40 cm high, its base area is 250 cm². The area of the larger speaker's base is 490 cm². Find the height and volume of the larger speaker? 40 cm. A = 250 cm² A = 490 cm²H = ? solution: 40 cm. A = 250 cm² We have the level 2 ratio 490 cm² to 250 cm². We take the square root to find the level 1 ratio. Volume = 56 cm × 490 cm² = 27 440 cm³ = = 490 250 Hh 75 H = 75 × 40 = 56 cm A = 490 cm²H = ? 8) Two right prisms with rectangular bases are similar. The base area of the smaller is 3.14 cm² and its height is 1.5 cm. The base area of the larger is 78.5 cm². What is the volume of the larger prism rounded to the nearest unit? Practice Exercises for Jaymesolution: Aa= 78.5 3.14 = 25 1 , therefore big V = 125 % (little V) = 589 cm3 9) Two cylindrical food cans are similar. The diameter of the smaller can equals the radius of the larger can. How many of the smaller cans could we fill with the full contents of the larger one? solution: We could fill 8 small cans with the contents of the larger one. 10) volume = 4800 cm³ 20 cm. volume = ? 5 cm. We're going to melt down the mother piece of metal to turn it into a bunch of little ones similar to the original. The measures are as shown. What is the volume of each smaller metal cylinder? solution: The level 1 ratio is 5 : 20 or ¼, so the level 3 ratio = (¼)³. The volume of each small piece will be 1/64 th the volume of the mother piece. 4800 ÷ 64 = 75 cm³. Practice Exercises for Jayme