The triangles below are similar. The ratio of the lengths of corresponding sides is 49 . a. What is the ratio (smaller to larger) of the perimeters? b. What is the ratio (smaller to larger) of the areas? The image is of two similar triangles with proportionate side of each marked as 4 in. and 9 in. respectively.

a.) 4 : 9

b.) 16 : 81

Explanation:

a.) Ratio of perimeter will still be

4 : 9

b.) The two similar triangles  ΔABC and ΔDEF as shown from the attached figures. That is

AB/DE = AC/DF = BC/EF

It is apparent that ΔAPB and ΔDEQ are also similar as all respective angles are equal. Hence,

AB/DE = AP/DQ = BP/EQ

Moreso ΔABC = 1/2 × BC × AP  and   ΔDEF = 1/2 × EF × DQ and

ΔAPB/ΔDEQ = BC/EF × AP/DQ

But 

AP/DQ = AB/DE = BC/EF

 and hence

ΔAPB/ΔDEQ = BC/EF × BC/EF

= BC^2/EF^2  and as

BC/EF = AC/DF = AB/DE

ΔAPB/ΔDEQ = AC^2/DF^2 = BC^2/EF^2

= AB^2/DE^2

Hence if sides of two similar triangles are in the ratio a:b, their areas are in the proportion a^2:b^2

As in given case sides are in the ratio of 4:9.

ratio of their areas is 4^2:9^2 or 16:81.

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