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geometry: similar triangles

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by buoyant » Wed Apr 23, 2014 2:21 pm
Triangles ABC and DEF are similar. What percent of the area of triangle DEF is triangle ABC?

1) sides AB and DE connect similar angles and AB is 3 times DE
2) side AB=9

[spoiler]OA:A[/spoiler]

Source: Veritas question bank
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Wed Apr 23, 2014 3:52 pm
buoyant wrote:Triangles ABC and DEF are similar. What percent of the area of triangle DEF is triangle ABC?

1) sides AB and DE connect similar angles and AB is 3 times DE
2) side AB=9
IMPORTANT: Notice that the question does not ask us to find the actual area of either triangle. We need only find their relative areas.

Target question: What PERCENT of the area of triangle DEF is triangle ABC?

Given: ∆ABC and ∆DEF are similar.

Statement 1: sides AB and DE connect similar angles and AB is 3 times DE
This means that EACH side of ∆ABC is 3 times as long as its corresponding side of ∆DEF
So, the base of ∆ABC is 3 times as long as the corresponding base of ∆DEF.
Likewise, the height ∆ABC is 3 times the height of ∆DEF.
Since area = (1/2)(base)(height), we can see that the area of ∆ABC is 9 TIMES the area of ∆DEF.
In other words, the area of ∆ABC is 900 percent the area of ∆DEF.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: side AB=9
This only tells us about one triangle.
Since there's no information about ∆DEF, we have no idea about the relative areas of the two triangles.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

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Brent
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by buoyant » Wed Apr 23, 2014 11:34 pm
Hi Brent,
nice approach.

I have a couple of questions here.

- what does "AB and DE connect similar angles" mean? does this mean they are corresponding sides of triangles ABC & DEF ?

- question says triangles are similar. but how do we know all other sides will have the same ratio, provided only one side ratio is given? why does it also include height?

we know that if each pair of corresponding sides of two triangles has the same ratio, we call these similar. In this case, we know that two triangles are similar and that the the ratio of one pair of corresponding sides is 3:1. Then how do we conclude about all other sides including height?

please explain.
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by GMATGuruNY » Thu Apr 24, 2014 12:30 pm
buoyant wrote:Hi Brent,
nice approach.

I have a couple of questions here.

- what does "AB and DE connect similar angles" mean? does this mean they are corresponding sides of triangles ABC & DEF ?
Correct.
The statement implies that AB and DE are corresponding sides.
While similar triangles do appear on the GMAT, the wording of this problem does not seem reflective of an actual GMAT problem.
The GMAT tends to avoid the term similar.
Generally, the GMAT finds a way to convey that two triangles are similar without using the actual term.
Rather than worry about the wording here, just be sure to understand the CONCEPTS behind this problem.
- question says triangles are similar. but how do we know all other sides will have the same ratio, provided only one side ratio is given? why does it also include height?

we know that if each pair of corresponding sides of two triangles has the same ratio, we call these similar. In this case, we know that two triangles are similar and that the the ratio of one pair of corresponding sides is 3:1. Then how do we conclude about all other sides including height?
When triangles are similar, ALL corresponding lengths -- including corresponding HEIGHTS -- are in the SAME RATIO.
Since AB = 3(DE), the height of ∆ABC must be 3 times the height of ∆DEF.
Check here for an OG13 problem that tests this concept:
https://www.beatthegmat.com/need-to-unde ... 61769.html
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by buoyant » Thu Apr 24, 2014 12:50 pm
Thanks Mitch!

i don't get one thing.
If in two similar triangles all angles are equal & all corresponding sides are in same ratio, then why the rules to recognize two similar triangles say side-angle-side (two corresponding sides in same ratio & the corresponding angles contained by these sides are equal) , angle-angle-angle (all corresponding angles are equal) , or side -side -side (all corresponding sides are in same ratio) ?
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by GMATGuruNY » Fri Apr 25, 2014 10:15 am
buoyant wrote:Thanks Mitch!

i don't get one thing.
If in two similar triangles all angles are equal & all corresponding sides are in same ratio, then why the rules to recognize two similar triangles say side-angle-side (two corresponding sides in same ratio & the corresponding angles contained by these sides are equal) , angle-angle-angle (all corresponding angles are equal) , or side -side -side (all corresponding sides are in same ratio) ?
SAS, AA, and SSS are 3 ways to prove that two triangles are similar.
If two triangles are shown to be similar, then all corresponding angles are equal, and all corresponding sides are in the same ratio.
The following sites explain SAS, AA, and SSS:
https://www.mathsisfun.com/geometry/tria ... nding.html
https://www.mathwarehouse.com/geometry/s ... eorems.php
https://www.mathopenref.com/similartriangles.html
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by buoyant » Fri Apr 25, 2014 3:20 pm
GMATGuruNY wrote:
buoyant wrote:Thanks Mitch!

i don't get one thing.
If in two similar triangles all angles are equal & all corresponding sides are in same ratio, then why the rules to recognize two similar triangles say side-angle-side (two corresponding sides in same ratio & the corresponding angles contained by these sides are equal) , angle-angle-angle (all corresponding angles are equal) , or side -side -side (all corresponding sides are in same ratio) ?
SAS, AA, and SSS are 3 ways to prove that two triangles are similar.
If two triangles are shown to be similar, then all corresponding angles are equal, and all corresponding sides are in the same ratio.
The following sites explain SAS, AA, and SSS:
https://www.mathsisfun.com/geometry/tria ... nding.html
https://www.mathwarehouse.com/geometry/s ... eorems.php
https://www.mathopenref.com/similartriangles.html


so, that means :
-> if all corresponding sides are in the same ratio => the triangles are similar => hence all the angles are equal and all sides have same ratio.
-> if all angles are equal => the triangles are similar => hence all the angles are equal and all sides have same ratio.
-> if 2 corresponding sides have same ratio + the contained corresponding angles are same => the triangles are similar => hence all the angles are equal and all sides have same ratio.

am i correct Mitch?
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