Start by finding the scale factor between the ornaments.
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Practice makes perfect
We are given that two ornaments are in the shape of similar triangles. We are asked to find the area of the larger ornament knowing that the area of the smaller ornament is 22.5 square centimeters. 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 ornament, we need to multiply the area of the smaller ornament, by the scale factor raised to the second power.
Area of the larger ornament
= 22.5( Scale factor)^2 [0.3em]However, we do not know the scale factor. To find it, we can use the perimeters of the ornaments. We will start by recalling 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 ornament is equal to the perimeter of the smaller ornament times the scale factor. We know that the perimeter of the larger ornament is 12 inches and the perimeter of the smaller ornament is 9 inches.
12 = 9 ( Scale factor) [0.3em]
We can solve this equation to find the scale factor.
Therefore, the scale factor is equal to 43. Now we can substitute this value into the first equation.
Area of the larger ornament
= 22.5( 4/3)^2
Finally, we will calculate the area of the larger ornament.
