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Geometry and Angles (My weakness)

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Geometry and Angles (My weakness)

by sanchu » Tue Feb 25, 2014 4:25 pm

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13th Edition Question Number 92.

A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above ground, how many feet above the ground does the ladder reach?

The answer set mentions that this is a 30-60-90 triangle. How would I have deduced this part? I visualized correctly, but I didn't know how to solve.
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by GMATGuruNY » Tue Feb 25, 2014 7:34 pm
sanchu wrote:13th Edition Question Number 92.

A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above ground, how many feet above the ground does the ladder reach?

The answer set mentions that this is a 30-60-90 triangle. How would I have deduced this part? I visualized correctly, but I didn't know how to solve.
Image

When ladder BC is elevated to an angle of 60 degrees, the result is ∆ABC.
Since AB is perpendicular to the ground, ∠BAC is a right angle.
Thus, ∆ABC is a 30-60-90 triangle.

In a 30-60-90 triangle, the sides are in the following ratio:
s : s√3 : 2s.
In ∆ABC, 2s = 70, implying that s=35.
Thus, the length of AB -- the middle side -- is 35√3.

Since the base of the ladder -- point C -- is 7 feet off the ground, the total distance between point B and the ground = [spoiler]35√3 + 7[/spoiler].
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by [email protected] » Tue Feb 25, 2014 9:07 pm
Hi sanchu,

Most geometry questions on the GMAT can be made easier to solve IF you draw a picture and fill in whatever information that you have. It's also worth noting that every question on the GMAT is very carefully designed, so you need to pay attention to the details and think about what those details remind you of.

In this question, the prompt tells us that a ladder is elevated to 60 DEGREES. Now WHY do you suppose that is? It's probably because this question is testing our knowledge of a 30-60-90 triangle (which is common enough on the GMAT that you're likely to see it).

So, try drawing a 30-60-90 triangle with a hypotenuse of 70 feet. If you know the rules of a 30-60-90 triangle, then you can calculate the other sides. Don't forget that the ladder is already 7 feet above ground, so that will factor into the solution.

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by mcdesty » Thu Jul 10, 2014 9:09 am
Your scratch paper should look like the attachment below.
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Ladder.jpg
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by Jeff@TargetTestPrep » Thu May 21, 2015 10:27 am
sanchu wrote:13th Edition Question Number 92.

A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above ground, how many feet above the ground does the ladder reach?

The answer set mentions that this is a 30-60-90 triangle. How would I have deduced this part? I visualized correctly, but I didn't know how to solve.
A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many feet above the ground does the ladder reach?

A/ 35
B/ 42
C/ 35√3
D/ 7 + 35√3
E/ 7 + 42√3

Solution:

We are given that the ladder of a fire truck is elevated to an angle 60 degrees above the ground and that the ladder has a length of 70 feet. We are also given that the ladder is 7 feet above the ground. The best thing to do in this situation is to draw a diagram.

Image

Notice that the resulting triangle in the sketch is a 30-60-90 right triangle. The ratio of the sides of a 30-60-90 right triangle is x : x√3 : 2x. We see that the hypotenuse length of 70 feet is equal to the "2x" from the 30-60-90 ratio. Thus, we can set up an equation and solve for x.

70 = 2x

x = 35

Because x = 35, we know that the side opposite the 60-degree angle or, in this case, the height of the ladder, is 35√3. The height of the ladder is 35√3 feet and the base of the ladder is 7 feet above the ground; thus, we know that the ladder reaches a total height above the ground of 35√3 + 7 feet.

The Answer is D

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by outsidethesidelines » Fri May 22, 2015 6:34 am
I worked through this problem yesterday and was conceptually fine on it -- i.e. recognized the 30-60-90 triangle, the fixed relationships for side lengths, and the need to add the 7 feet to the final answer -- but I ended up choosing the wrong answer because I didn't assign the 30 and 60 degree angles correctly, and obviously that gave me the wrong answer when I calculated the lengths via the fixed ratios for 30-60-90 triangles.

Reading back over this, the prompt seems to be really vague about where 60 degree angle is formed (i.e. between the ladder and the truck, or between the ladder and the building). Am I just misreading it when I think of it that way?
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by [email protected] » Fri May 22, 2015 9:45 am
Hi outsidethelines,

Since the prompt tells us that the LADDER is elevated to an angle of 60 degrees, that angle is relative to the flat ground (or in this case, "parallel" to the ground).

It often helps to draw pictures when dealing with Geometry questions. Beyond the obvious utility (you can then put in the numbers and "see" how everything relates, you can translate wordier questions 'step by step' and not be overwhelmed by the information.

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by vickysan » Wed Jun 03, 2015 9:20 pm
mcdesty wrote:Your scratch paper should look like the attachment below.
Great visual. Thanks for providing the scratchwork, and Rich, thank you for the insight as well.
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by Brent@GMATPrepNow » Fri Sep 07, 2018 1:14 pm
sanchu wrote: A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above ground, how many feet above the ground does the ladder reach?
A) 35
B) 42
C) 35√3
D) 7 + 35√3
E) 7 + 42√3
Here's an idea of what's going on....
Image
When we compare the big triangle with the BASE 30-60-90 special triangle (which you must memorize for test day!), we can see that the hypotenuse of the big triangle is 35 times as long as the hypotenuse of the base triangle.
This means the big triangle is 35 times the size of the base triangle.

This means that, on the big triangle, the side opposite the 60 degree angle must be 35√3
Image

Now before we (incorrectly) choose answer choice C, we must keep in mind that the question tells us the base of the ladder is 7 feet above the ground...
Image

So, the TOTAL distance from the top of the ladder to the ground = 35√3 + 7

Answer: D

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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