Question: Tilted Triangle Area

Comment on Tilted Triangle Area

Why wouldn't the answer be D? In your example, you extended the base which means it's no longer 5 in order to make angle ACB 90 degrees. You have also mentioned that theses pictures are not drawn to scale; so why couldn't it be understood that angle ACB was already 90 degrees, especially since we knew angle ABC was 60 degrees?
greenlight-admin's picture

If angle ACB were already 90 degrees, then the triangle would already be a 30-60-90 special right triangle. HOWEVER, its given sides of length 5 and 12 would not conform to the known lengths of 30-60-90 special right triangles. This means we can conclude that triangle ABC is NOT a 30-60-90 special right triangle.

So when you extended the base to determine the height, did the base number not change from being 5; therefore, causing the area to change all together?
greenlight-admin's picture

That's correct. We just extend an imaginary line (indicated by a dotted line) along the base so we can determine the triangle's height. If we change the length of side CB (as you suggest) then we are no longer dealing with the same triangle we started with.

Okay, and sorry again for the redundancy in responses. When I was doing this problem I did notice that the bases did not fit the 30-60-90 triangle rule. So I resulted to the Pythagorean Theorem and squared the given sides and after adding them got 169. So I thought 13 had to be the height and multiplied it by 5 (divided my answer by 2) and got 32.5. So I was wondering why that approach did not lead me to the correct answer.

I thought the height had to be greater than 6(times the square root of 3), since the base was less than 6. So I thought (like exactly what you said), extending the base would have changed the triangle we started with, and went the route I went as mentioned earlier.

I am truly appreciative of your help and these resources.
Thank you again
greenlight-admin's picture

The Pythagorean theorem only applies to right triangles. Since we are not told this is a right triangle, we can't use that theorem.

Thanks for your persistence in asking these questions, I am sure many of us made the same mistake!

So when you extended the base to determine the height, did the base number not change from being 5; therefore, causing the area to change all together?

You stated in your response that the given triangle ABC was not a 30-60-90 special right triangle until you made it so by extending the base when looking for the height (since the sides of the lengths given did not conform until you extended the base).

So this is why I'm not understanding; for if the base had to be extended to find the height, to make a right 30-60-90 triangle (because what was given in the question was not a 30-60-90 right triangle); than, that means the base could no longer be 5 (but 6), since what was given had to changed via the extension of the base; to find the height (to therefore solve for the area).

Sorry for the beginning repeat, I wasn't finished with my initial comment before it was sent prematurely. Thank you for your help by the way; I just want to do well and am appreciative of the time spent and dedicated to these resources.

Thank you again
greenlight-admin's picture

We are creating a hypothetical 30-60-90 right triangle by extending the base (using an imaginary line). We use that hypothetical 30-60-90 right triangle to find the height of the original ABC triangle.

lol No prob Jimmy, and Thank you so much again for all your help Brent, you are truly appreciated. Shout out to Chris C and his awesome programming skills :D

What is enlargement factor?when it is used?
greenlight-admin's picture

The "enlargement factor" is a number representing how much larger one triangle is when compared with another triangle. We use it when comparing similar triangles.

More here: https://www.greenlighttestprep.com/module/gre-geometry/video/870

And here: https://www.greenlighttestprep.com/module/gre-geometry/video/872

Hi, I just have two question. we figured out that the "enlargement factor" = 6, right?
so, how come the base is 5, should n't be 6 to follow the special triangle 30-60-90
12/2= h/ root 3= 6/1? not 5 right? I hope u got me.
2nd question is when should use the fact that in similar triangles the ratio of any pair of corresponding sides is the same and when we should use "enlargement factor"?
greenlight-admin's picture

Question 1: The side with length 5 is NOT part of a special triangle 30-60-90 triangle.

Once we CREATE a special triangle 30-60-90 triangle (by extending a perpendicular line from A down to the base), the 30-60-90 triangle has a base that's LONGER than 5.

Once we CREATE the special triangle 30-60-90 triangle, the only side we know about is the hypotenuse, which has length 12.

Question: Any time we have two similar triangles, it may be useful to apple the enlargement factor technique.

The formula for finding the area ofa triangle,which module was it given?
greenlight-admin's picture

It's in the Geometry module, here: https://www.greenlighttestprep.com/module/gre-geometry/video/865

Cheers,
Brent

Hi Brent,

Why can't we use the pythagorean theorem to find the height. Height ends up being root 119, area is 5*root 119 / 2 choice C?
greenlight-admin's picture

Hi aferro1989,

That's a good idea, but there's one problem.
At 0:36 in the video, we create a blue right triangle.
The hypotenuse of this right triangle has length 12, BUT the length of the bottom leg is NOT 5.
The length of the bottom leg is 5 PLUS some unknown value.
Since we don't know the length of the bottom leg, we can't apply the Pythagorean Theorem.

Does that help?

Cheers,
Brent

Thanks Brent! Still a bit confused - conceptually it seems like we are creating an artificial triangle with the blue line to determine the height, but aren't we still using this artificial triangle when we say the resulting triangle is a 30-60-90 triangle and use the 30-60-90 properties to determine the height and resulting area?
greenlight-admin's picture

Yes, we are using the blue (artificial) triangle to determine the height (and resulting area).

This is fine, because we are certain that the triangle is a 30-60-90 triangle, AND we are certain that the hypotenuse has length 12.

Given this, we have enough information to find the height of the blue (artificial) triangle.

That said, your strategy (of applying the Pythagorean Theorem) WOULD be valid IF we had more information.

Aside: if we want to use the Pythagorean Theorem to find the length of one side, we must know the lengths of the other TWO sides.

In the case of your proposed solution, all we know is that the hypotenuse has length 12. We don't know the length of the bottom leg of the triangle.

Here's an image showing the missing measurement: https://imgur.com/j9fwDDy

Does that help?

Cheers,
Brent

That's helpful, thank you Brent!

Thanks Brent for all your help. Please my question is about the imaginary height, does it mean if the base was extended from 5 the height will remain the same?
greenlight-admin's picture

To calculate the area of any triangle, we can make any side the base.
For each side we use as base, we'll get a different height.

To see this in action, watch 4:50 to 6:00 of the video on finding the area of a triangle (https://www.greenlighttestprep.com/module/gre-geometry/video/865)

Cheers,
Brent

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