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If two triangles are similar and the ratio of their corresponding sides is 4 : 1, what is the ratio of their areas?
4:1
2:1
8:1
16:1
16:1
Whenever the ratio of sides is given as a:b, the ratio of areas is simply the square of the ratio, a2:b2. Simply compute 42:12 to arrive at the answer in 5 seconds.
Two triangles are similar. The ratio of their corresponding sides is 4 : 1.
Area2тАЛArea1тАЛтАЛ=(Side2тАЛSide1тАЛтАЛ)2
Whenever the ratio of sides is given as a:b, the ratio of areas is simply the square of the ratio, a2:b2. Simply compute 42:12 to arrive at the answer in 5 seconds.
Students often make the mistake of assuming the ratio of areas is the same as the ratio of sides (4:1) or mistakenly doubling the ratio (8:2) instead of squaring it.
State the Area Property
For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
Area2тАЛArea1тАЛтАЛ=(S2тАЛS1тАЛтАЛ)2
Substitute the Given Values
The ratio of the corresponding sides is given as S2тАЛS1тАЛтАЛ=14тАЛ. Substituting this into the formula:
Area2тАЛArea1тАЛтАЛ=(14тАЛ)2
Perform the Calculation
Squaring the ratio yields 1242тАЛ=116тАЛ.
116тАЛ=16:1
D is correct because the area ratio is the square of the side ratio, resulting in 42:12=16:1.
This property also applies to the volumes of similar 3D solids, where the ratio of volumes is the cube of the ratio of corresponding linear dimensions (e.g., sides or radii).
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