Ratio of Sides and Ratio of AreasDate: 02/11/99 at 00:45:40 From: Danny Bradley Subject: High school Geometry The corresponding sides of two similar triangles are in the ratio of 1:7. What is the ratio of their areas? Date: 02/11/99 at 18:17:50 From: Doctor Pat Subject: Re: High school Geometry Danny, When two triangles are similar, if a pair of corresponding sides is in the ratio of 1:7 then every other pair of corresponding LENGTHS will be in the ratio 1:7. The bases are 1:7, the heights are 1:7, etc. Now let's find the areas. Call one triangle's area A and the other triangle's area a. Call the base of the larger triangle B, and the smaller triangle's base b. Call the height of the larger triangle H, and the smaller triangle's height h. So the area of the big triangle A = .5 B H, and the area of the smaller triangle is a = .5 b h. Now B = 7b and H = 7h, so A =.5(7b*7h) = .5 49 bh That makes the ratio A .5 * 49 * b h --- = ------------- a .5 * b h since the .5 bh all cancels out, we are left with A/a = 49:1. You might notice that this is the square of the ratio of the lengths. You can convince yourself that this always works by drawing two squares (all squares are similar) with different sides. The ratio of the areas is ALWAYS the square of the ratios of the sides. Good luck, - Doctor Pat, The Math Forum http://mathforum.org/dr.math/ |
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