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The ratio of the lengths of two corresponding sides of two similar triangles is 17 : 13. The ratio of the areas of these two triangles, in the order mentioned, is:
289 : 169
17 : 13
1717тАЛ:13
290 : 170
289 : 169
For similar triangles, the ratio of areas is always the square of the ratio of their corresponding sides. Simply square both parts of the given ratio: 172:132=289:169.
Ratio of the lengths of corresponding sides of two similar triangles = 17 : 13
Area2тАЛArea1тАЛтАЛ=(side2тАЛside1тАЛтАЛ)2
For similar triangles, the ratio of areas is always the square of the ratio of their corresponding sides. Simply square both parts of the given ratio: 172:132=289:169.
Students often mistakenly assume the ratio of areas is the same as the ratio of sides, forgetting to square the values.
State the Geometric Property
For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
A2тАЛA1тАЛтАЛ=(S2тАЛS1тАЛтАЛ)2
Substitute Given Ratio
Given the side ratio S1тАЛ:S2тАЛ=17:13, substitute these into the area formula.
A2тАЛA1тАЛтАЛ=(1317тАЛ)2
Perform Calculation
Calculate the squares: 172=289 and 132=169.
A2тАЛA1тАЛтАЛ=169289тАЛ
A is correct because the ratio of areas of two similar triangles is the square of the ratio of their corresponding sides, which is 172:132=289:169.
This property also applies to 3D similar shapes (like spheres or cubes), where the ratio of volumes is the cube of the ratio of the sides.
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