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The ratio of the lengths of two corresponding sides of two similar triangles is 5 : 3. The ratio of the areas of these two triangles is:
5 : 3
10 : 6
25 : 9
125 : 27
25 : 9
If the side ratio is a:b, the area ratio is simply a2:b2. Just square both terms of the given ratio.
Ratio of lengths of corresponding sides of two similar triangles = 5 : 3.
Area2тАЛArea1тАЛтАЛ=(Side2тАЛSide1тАЛтАЛ)2
If the side ratio is a:b, the area ratio is simply a2:b2. Just square both terms of the given ratio.
Students often incorrectly assume the area ratio is the same as the side ratio or cube the ratio as they would for volume.
Identify the Property of Similar Triangles
According to the Area Theorem for similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
Area2тАЛArea1тАЛтАЛ=(Side2тАЛSide1тАЛтАЛ)2
Substitute the Given Values
Given the side ratio is 5:3, we substitute Side2тАЛSide1тАЛтАЛ=35тАЛ into the formula.
Area2тАЛArea1тАЛтАЛ=(35тАЛ)2
Calculate the Result
Square both the numerator and the denominator: 52=25 and 32=9.
Area2тАЛArea1тАЛтАЛ=925тАЛ
C is correct because squaring the ratio of sides 5:3 results in 25:9.
This concept is a special case of the scale factor principle; for similar 3D solids, the ratio of volumes is the cube of the side ratio (a3:b3).
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