We are given similar polygons DPQR~DKLM. Also PQ=4,KL=6 with the area of DPQR 12 square cm. We are asked to find the area of DPQR.
Since the polygons are similar their corresponding sides are in proportion. Thus:
In particular, the scale factor is this ratio or `"PQ"/"KL"=4/6=2/3 `
For similar figures, all corresponding linear measurements will be in the same ratio as the scale factor while all corresponding area measurements will be in the ratio of the square of the scale factor.
For these figures, the ratio of areas will be `2^2:3^2=4:9 `
Since the area of DPQR is 12, we can set up a proportion to find the other area:
`12/A=4/9 ==> A=27 `
The area of DKLM is 27 square cm
12 x 9/4=27
In a situation of similar polygons, the ratio of sides leads us to the ratio of areas. Since similar sides have a ratio of 6/4 wich simplifies to 3/2, the ratio of areas is 3^2/2^2 or 9/4.
Multiplying the area of the smaller polygon by 9/4 gives us the area if the larger polygon.
So 12 x 9/4 = 108/4 = 27 square cm.