Start by finding the scale factor between the pieces of wood.
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Practice makes perfect
We are given that two rectangular pieces of wood are similar. We are asked to find the area of the larger piece knowing that the area of the smaller piece is 12 square inches. Let's start by recalling the relationship between the areas of similar figures.
Area of Similar Figures
If figure B is similar to figure A by a scale factor, then the area of B is equal to the area of A times the square of the scale factor.
Therefore, to get the area of the larger piece, we need to multiply the area of the smaller piece, by the scale factor raised to the second power.
Area of the larger piece
= 12( Scale factor)^2 [0.3em]However, we do not know the scale factor. To find it, we can use the ratio of the perimeters of the two pieces. Note that the ratio is less than 1.
2:3 = 2/3 < 1
Therefore, 23 is a ratio of the perimeter of the smaller piece to the
perimeter of the larger piece. For simplicity, let s represent the perimeter of the smaller piece and let l represent the perimeter of the larger piece.
2/3 = s/l
Now let's recall the relationship between the perimeters of similar figures.
Perimeter of Similar Figures
If figure B is similar to figure A by a scale factor, then the perimeter of B is equal to the perimeter of A times the scale factor.
According to this rule, the perimeter of the larger piece is equal to the perimeter of the smaller piece times the scale factor.
l =
s( Scale factor)
We can rewrite this equation and use the ratio to get the scale factor.
Therefore, the scale factor is equal to 32. Now we can substitute this value into the first equation.
Area of the larger piece
= 12( 3/2)^2
Finally, we will calculate the area of the larger piece.
