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тЦ│ ABC is such that AB = 5 cm, BC = 4 cm and CA = 4.5 cm. тЦ│ XYZ is similar to тЦ│ ABC. If YZ = 8 cm, then find the perimeter of тЦ│ XYZ.
28┬аcm
27┬аcm
26┬аcm
25┬аcm
27┬аcm
Calculate the perimeter of the base triangle (5 + 4 + 4.5 = 13.5) and multiply by the ratio of corresponding sides (8/4 = 2). Thus, 13.5 * 2 = 27 cm.
AB = 5 cm, BC = 4 cm, CA = 4.5 cm are sides of triangle ABC. Triangle XYZ is similar to ABC and YZ = 8 cm.
Perimeter(╬ФABC)Perimeter(╬ФXYZ)тАЛ=BCYZтАЛ
Calculate the perimeter of the base triangle (5 + 4 + 4.5 = 13.5) and multiply by the ratio of corresponding sides (8/4 = 2). Thus, 13.5 * 2 = 27 cm.
Students often fail to identify the corresponding sides correctly, mistaking AB or AC for the side corresponding to YZ, leading to incorrect ratio calculations.
Calculate the perimeter of triangle ABC
The perimeter is the sum of all sides: 5+4+4.5.
PABCтАЛ=5+4+4.5=13.5┬аcm
Determine the scale factor
Since ╬ФXYZтИ╝╬ФABC, the ratio of corresponding sides is BCYZтАЛ=48тАЛ=2.
k=48тАЛ=2
Calculate the perimeter of triangle XYZ
The ratio of perimeters of similar triangles equals the ratio of their corresponding sides: PXYZтАЛ=PABCтАЛ├Чk.
PXYZтАЛ=13.5├Ч2=27┬аcm
B is correct because the perimeter of ╬ФABC is 13.5┬аcm and the similarity ratio is 2, resulting in a perimeter of 27┬аcm for ╬ФXYZ.
This concept of ratios in similar shapes applies directly to areas as well, where the ratio of areas is the square of the ratio of sides.
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