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Please help, I've been working on this problem for ages and I can't seem to get the answer.

The heights of two similar triangles are in the ratio 2:5. If the area of the larger triangle is 400 square units, what is the area of the smaller triangle?

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2 Answers 2

Hint: When you scale the linear dimensions of a figure by the factor $\lambda$, the area gets scaled by the factor $\lambda^2$. To get from the big triangle to the small one, you scale linear dimensions by the factor $\lambda=\frac{2}{5}$.

Or, in a version I like much less, take a triangle with base $b$, height with respect to that base $h$. Then new base, new height are $\lambda b$, $\lambda h$. So old area is $\frac{1}{2}bh$, new area is $\frac{1}{2}(\lambda b)(\lambda h)=\lambda^2 \frac{1}{2}bh$.

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It may help you to view your ratio as a fraction in this case. Right now your ratio is for one-dimensional measurements, like height, so if you were to calculate the height of the large triangle based on the height of the small triangle being (for example) 3, you would write:

$3 \times \frac52 =$ height of the large triangle

Or, to go the other way (knowing the height of the large triangle to be, say, 7) you would write:

$7 \times \frac25 = $ height of the small triangle

But this is for single-dimensional measurements. For a two-dimensional measurement like area, simply square the ratio (also called the scalar):

area of small triangle $\times (\frac52)^2 =$ area of large triangle.

This can be extended to three-dimensional measurements by cubing the ratio/fraction.

(you'll know which fraction to use because one increases the quantity while the other decreases. So if you find the area of the large triangle to be smaller than the small triangle, you've used the wrong one!)

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