scale factors and similarity: DA=72, AE=36, EB=24, area ABC=1950; find area of ABC

Simonsky

Junior Member
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Jul 4, 2017
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128
Hello,

I've been staring at a question for ages to do with congruency and similarity as well as scale factors and parallel lines-can't seem to get anywhere.

The question is no. 10 on the attachment -I hope it can be seen clearly.

For any pointers-thanks!Scale factorscongruencysimilarity.jpg
 
Hello,

I've been staring at a question for ages to do with congruency and similarity as well as scale factors and parallel lines-can't seem to get anywhere.

The question is no. 10 on the attachment -I hope it can be seen clearly.

For any pointers-thanks!

Can you identify a pair of similar triangles (one being ABD, to start with)?
 
It is not hard to solve for a

For b, since you know the length of two sides and the area of triangle DAB, you can use Heron's formula to find lengths of DB, DF, and FB. You will also know the angle ADB, which is the same as angle DBC. Since you know angle DBC and the lengths of FB and CB, you will know the length of CF. Then you can find the length of AF. Now all the lengths of edges of triangle ABC are known, you can find the area of it.
 
For b, since you know the length of two sides and the area of triangle DAB, you can use Heron's formula to find lengths of DB, DF, and FB. You will also know the angle ADB, which is the same as angle DBC. Since you know angle DBC and the lengths of FB and CB, you will know the length of CF. Then you can find the length of AF. Now all the lengths of edges of triangle ABC are known, you can find the area of it.

It can be done much more simply than that. Once you have the lengths from part (a), you just need the fact that triangles on the same base have areas proportional to their altitudes, plus a little more with similar triangles if necessary.
 
Great job. You have my solute.

It can be done much more simply than that. Once you have the lengths from part (a), you just need the fact that triangles on the same base have areas proportional to their altitudes, plus a little more with similar triangles if necessary.

The heights of similar triangles are proportional.
 
The heights of similar triangles are proportional.

Yes, that's the second key idea. Although ABC and ABD are not themselves similar, their altitudes are also the altitudes of other triangles that are similar, so we can find their ratio. (I said "if necessary" because there are other ways to think of it.)
 
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