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.
1 answer:
Answer:
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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