If the ratio of the sides of two similar triangles is 4:9, how do you find the ratio of their areas?

2 Answers
Dec 22, 2016

16:8116:81

Explanation:

Scale factor for the sides of these triangles. k =4/9.k=49.

Therefore the ratio of area will be:

k^2 = "Area Triangle A"/"Area triangle B"k2=Area Triangle AArea triangle B

k^2 = (4/9)^2k2=(49)2

= 16/81=1681

Dec 22, 2016

Ratio of their areas is 16:8116:81.

Explanation:

Let us have two similar triangles DeltaABC and DeltaDEF as shown below. As they are similar, we have

(AB)/(DE)=(AC)/(DF)=(BC)/(EF)

Let us also draw perpendiculars AP and DQ from A and D respectively on to BC and EF as shown.
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It is apparent that DeltaAPB and DeltaDEQ are also similar as all respective angles are equal. Hence,

(AB)/(DE)=(AP)/(DQ)=(BP)/(EQ)

We also have DeltaABC=1/2xxBCxxAP and DeltaDEF=1/2xxEFxxDQ and

(DeltaAPB)/(DeltaDEQ)=(BCxxAP)/(EFxxDQ)=(BC)/(EF)xx(AP)/(DQ)

But (AP)/(DQ)=(AB)/(DE)=(BC)/(EF) and hence

(DeltaAPB)/(DeltaDEQ)=(BC)/(EF)xx(BC)/(EF)=(BC^2)/(EF^2) and as

(BC)/(EF)=(AC)/(DF)=(AB)/(DE)

(DeltaAPB)/(DeltaDEQ)=(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.