A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

(A) 19,200
(B) 19,600
(C) 20,000
(D) 20,400
(E) 20,800

This is a very basic GMAT geometry question: area of a rectangle given the perimeter and diagonal. System of equations. Again, the setup is easy peasy. It gets a bit tricky once you need to simplify to an answer. Depending on how you approach the execution you might get stuck. In addition, the answer choices are bunched closely together so you can't really estimate/approximate.

What should you do if encountered with this type of situation on your GMAT? Panic! Just kidding. Remind yourself that there's likely a simple way to approach things. Remember that unless a number is prime you can pull it apart. Often then there's an easier "shape" to a number depending on what you're trying to do.

In this case, by pulling out some factors we can:

  1. Do easier math with known multiples
  2. Divide by two

Also, with exponents often you end up factoring. So given that you have a bunch of exponents present factoring should come to mind. These simple setup/tricky execution questions aren't uncommon so it's important to work on being flexible in how you approach your GMAT arithmetic. Hope this is helpful. Comment with any questions!

A Small Rectangular Park has a perimeter of 560 GMAT Solution

In the figure above, equilateral triangle ABC is inscribed in the circle. If the length of arc ABC is 24, what is the approximate diameter of the circle?

(A) 5
(B) 8
(C) 11
(D) 15
(E) 19

It's key to know that an equilateral triangle has equal sides and equal angles measuring 60 degrees each. So each angle of an equilateral triangle inscribed in a circle cuts off 1/3 of the circle. Why is that? Review the basics! An angle on the circumference of a circle cuts off twice that angle's measure. So a 60 degree angle on the edge of a circle cuts off 120 degrees or 1/3 of a circle (120/360 degrees). A 60 degree angle in the center of a circle only cuts off 60 degrees (it doesn't have the extra space to open up). Examples below to illustrate this. Also, there's a special case for triangles that have the diameter of a circle as their base. Because the diameter is a 180 degree central angle, the angle opposite on the circumference of the circle must be 90 degrees. Not relevant for this question but good to know.

In the figure above, equilateral triangle ABC is inscribed in the circle.

How much of the circle is arc ABC? It's covering the same part of the circumference as two of the 60 degree angles from triangle ABA. Each of those angles represent 1/3 of the circle so arc ABC is 2/3 of the circle. If 2/3 is 24, 1/3 of the circle's circumference is 12, and the entire circumference is 36. So Pi*diameter = circumference. Pi*D = 36. D = 36/Pi. D = 36/3.14. A little less than 12. 11 is by far the closest answer choice. A little rusty feeling on these GMAT geometry rules? The rules are the easy part of GMAT studying. Get your fundamentals in shape ASAP. Avoid churning through questions before getting your basics GMAT quant organized. You don't want to waste good material and it's much harder to focus on the basics while working on tougher/trickier GMAT questions. Comment with any questions or additions. Happy studies!

In the figure above equilateral triangle ABC is inscribed in the circle. If the length of arc ABC is 24, what is the approximate diameter of the circle?

A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What is the ratio of the height of the cone to the radius of the hemisphere?

(A) √3:1
(B) 1:1
(C) 1/2/1
(D) √2:1
(E) 2:1

 

In the rectangular coordinate system above, the area of triangular region PQR is

In the rectangular coordinate system above, the area of triangular region PQR is

(A) 12.5

(B) 14

(C) 10√2

(D) 16

(E) 25

For GMAT geometry questions be active with diagrams. Don't be satisfied with what you're given. Try to make simple additions that provide more information (inferences). In general when trying to find an area on the coordinate plane you need to make a square, rectangle, or at the least, a right triangle. Right angles make measurement possible. Also, be open to the idea of adding/subtracting the areas of two or more shapes in order to solve for the desired area. In this case I'd draw two extra lines to make a rectangle.

In the rectangular coordinate system above, the area of triangular region PQR is

With that you should notice that you've created extra triangles and that those are right triangles. On GMAT geometry right triangles are your friends because you can do Pythagorean theorem and calculating are uses on the sides of the triangle (there's no need to drop an altitude). Now you can calculate the area of the rectangle and then subtract from that the areas of the three right triangles. That will leave the area of the middle triangle. Comment with any question or additions. Happy studies!

In the rectangular coordinate system above, the area of triangular region PQR is GMAT

To see it done live here is a video solution:

Which of the following lists the number of points at which a circle can intersect a triangle?

(A) 2 and 6 only

(B) 2, 4 and 6 only

(C) 1, 2, 3 and 6 only

(D) 1, 2, 3, 4 and 6 only

(E) 1, 2, 3, 4, 5 and 6 only

I'm often asked: "What's the most important thing to work on in order to succeed on the GMAT". Not a simple question. Of course you need all of the quant and verbal fundamentals. And you're going to need to learn GMAT specific strategy. Above all that though is what I think is the most important quality: organized curiosity. You have to be willing to explore a bit. Not randomly though. In an organized way. Conducting mini experiments to feel out a question. This geometry question from the GMAT prep tests is a good example of the type of GMAT puzzle that rewards those willing to try a few things. Most often students stare at this one and refuse to play a little bit drawing out scenarios. Time pressure gets the best of them and then it's over. You have to ignore the clock a bit so you can let some ideas flow. GMAT time pressure is real. In that it's a timed exam and for most people that adds a formidable constraint. Still, if you're going to spend precious time working on a question better make that quality time leaving yourself open to the type of thinking that tends to lead to success. Developing the habit of organized curiosity requires taking the risk of ignoring the clock. Hopefully you'll find that by opening things up a bit you think more flexibility and creatively and give yourself the time to organize things properly so that you actually are not only more effective but more efficient. This isn't a skill that you just turn on. It needs to be practiced. And while you're practicing you're probably going to fail a whole bunch. But remember to keep practicing how you actually want to perform. It's true that in the short term you might be able to get away with lower quality thinking and make some gains refining that but if you're looking to fundamentally change your GMAT performance you likely need to not only work on GMAT content by your general approach to problem solving. Diagram below with the solution and here's a video explanation: Which of the following lists the number of points at which a circle can intersect a triangle

Which of the following lists the number of points at which a circle can intersect a triangle

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