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The corresponding sides of two similar triangles are in the ratio of 3:4. If the area of the smaller triangle is 54 sq cm, then the area of the bigger triangle is:
72 sq cm
96 sq cm
81 sq cm
108 sq cm
96 sq cm
Since the side ratio is 3:4, the area ratio is 32:42=9:16. If 9 units = 54, then 1 unit = 6, and 16 units = 16├Ч6=96.
Ratio of corresponding sides of two similar triangles = 3:4. Area of smaller triangle = 54 sq cm.
Area2тАЛArea1тАЛтАЛ=(Side2тАЛSide1тАЛтАЛ)2
Since the side ratio is 3:4, the area ratio is 32:42=9:16. If 9 units = 54, then 1 unit = 6, and 16 units = 16├Ч6=96.
Students often multiply the area by the ratio 4/3 instead of squaring the ratio (4/3)2, leading to an incorrect result of 72.
State the Geometric Property
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
AreabigтАЛAreasmallтАЛтАЛ=(SidebigтАЛSidesmallтАЛтАЛ)2
Substitute Given Values
Substitute the ratio 3/4 and the area of the smaller triangle into the formula.
AreabigтАЛ54тАЛ=(43тАЛ)2=169тАЛ
Calculate the Area
Cross-multiply to solve for the area of the bigger triangle: AreabigтАЛ=54├Ч916тАЛ.
AreabigтАЛ=6├Ч16=96
B is correct because the ratio of areas is the square of the side ratio, 9:16, making the larger area 54├Ч916тАЛ=96 sq cm.
This property also applies to the ratio of volumes of similar 3D figures (e.g., spheres, cubes) where the ratio is the cube of the side ratio: (s1тАЛ/s2тАЛ)3.
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