Question: 3 Triangles

Comment on 3 Triangles

Why is the enlargement factor of the isosceles triangle not root18?
Root18 times root2 = 6.
gmat-admin's picture

The enlargement factor of the isosceles triangle can be written in several equivalent ways.
In the video, we say the enlargement factor = 6/√2, but we don't bother about simplifying that value, since we're not asked to find that specific value.
Rather, we're just going to use it to find the other lengths in the triangle.

In actuality, 6/√2 = 3√2 = √18. They all have the exact same value. Here's why:
Take: 6/√2
Multiply top and bottom by √2 to get: (6√2)/2
Simplify to get: 3√2
So, 6/√2 = 3√2

Now take: 3√2
Rewrite 3 as √9 to get: (√9)(√2)
Simplify to get: √18

So, the enlargement factor of the isosceles triangle = 6/√2 = 3√2 = √18

gmat-admin's picture

How would you then solve the enlargement factor for the second (30-60-90) triangle using "root 18" as side BD?

gmat-admin's picture

Side BC has length √18.
The corresponding side of the base 30-60-90 triangle has length 2
So, the enlargement factor = (√18)/2

If we wish, we can simplify (√18)/2

√18 = (√9)(√2) = 3√2
So, the enlargement factor = (3√2)/2

Please note that (√18)/2 = (3√2)/2 = 6/√2


Perfectly used 1x:1x:sqrt(2)x and 1x:sqrt(3)x:2x

Hi Brent, at video 2:29 for triange ABD, to clarify so we have all 3 sides as 6/√2?
gmat-admin's picture

We're told that AD = BD in the given information, which means triangle ABD is an isosceles triangle with TWO angles measuring 45 degrees each.

So, once we conclude that side BD = 6/√2, we can also conclude that side AD = 6/√2, since we are told that AD = BD in the given information.

However, since the given information also tells us that side AB = 6, we can't have the say that all three sides of triangle ABD have length 6/√2.

Great thanks Brent for your prompt reply. Understand now that 6/√2 is both the enlargement factor and same for as BD=AD but side AB stay same as 6? Thanks Brent
gmat-admin's picture

The length of side AB (the hypotenuse of the 45-45-90 triangle) is 6.
When we compare this length (6), √2 to the length of the hypotenuse in the BASE 45-45-90 triangle, we are able to conclude that the enlargement factor is 6/√2.

In the BASE 45-45-90 triangle, each leg has length 1.
So when we apply the enlargement factor to each leg, each leg now has a length of 6/√2.

Does that help?

Great explanation and crystal clear now that orinial AB value stay the same, thanks Brent

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