Hello, hello! Welcome to GMAT Tuesdays with Kevin, me!
Today we're gonna do a tour through geometry um if you come to the Magoosh GMAT blog a lot you
probably saw this challenge question that was on the side of the website and I want
to work through the problem with you so you can see how to solve it and the reason I'm
calling this a tour through geometry is because that's kind of what it is this type of question
is really just seeing how much of geometry do you really know. So first the question it says if a b
is a square with the area of 625 and c e f d is a rhombus with an area of 500 then the area
of the shaded region is and we have our options here. Um so important things to pay attention to
one figure not drawn to scale. That means it's not exactly like a square because I can't draw very
well and even on the test they will have this information there so you really don't want to
trust what you see you want to trust the numbers and your knowledge of geometry. Other things we
want to notice, um we're dealing with a square and we're told the area of it and we're dealing
with a rhombus and we're told the area of that. Um and then it's always good to kind of take a quick
glance at the answer choices just to see what is here. Um okay so it's going from smallest to
largest 125 at the lowest end 275 at the high end one thing you know sometimes you're running out
of time on the test or you want to catch up you're pacing yourself and you know you need to catch up
and move through a question quickly so you might have to guess. One thing that I would do to help my
chances of guessing with this type of problem is I can see 625 and 500 here and someone who's not
thinking very clearly or is rushing might think oh the area of this part here is just 625 minus
500 so it's gonna be 125. That is a trap that's a trap answer right there so you can immediately
get rid of that answer choice and then at least you're choosing from four instead of five answer
choices. Then maybe you can do some other stuff to eliminate those other possible answer choices. Okay
so with math problems it's always helpful for myself to kind of get a sense of what I'm going
to do I want to come up with like a road map or a plan for attacking the problem before I
actually start doing the problem one mistake that students often make with math is they
just start doing stuff without really having an idea of where they're going and that can
end up wasting a lot of time because you might go down the wrong path and then have to go back
to the beginning and start over. So ultimately if you spend just you know 15-20 seconds at
the beginning figuring out what you need to do and then doing it you'll probably save time in the end.
So ultimately I know I'm dealing with a square a rhombus a triangle and a rectangle. So
what I'm going to do here is basically I'm given the areas so I know I need to I'm
going to have to find the sides of these shapes I also know that I have right here if I draw a line
right here let me make that easy to see. Um I have a rectangle now and then a triangle and so if I
find the area of this rectangle and the area of this triangle and add them together then I'll know
what the area is of these shaded regions and so I have these mystery signs that I need to figure out
for the square and the rhombus but since I have the area here I know that I can do
it. So that's my plan and let's just dive in and do it so the area of the square is 625
and if you guys remember what the area of the square is it's the length times the width. Um so
that's going to be 25 times 25 so now I know okay I've got a 25 length there 25 there 25 here
awesome um so I'm pretty close to actually solving the area of this rectangle here. I just need to
find out what that oh that's really ugly let's try that again bam. Okay, so I need to find the
error or the length of this part right here. I know the whole thing is 25 if I knew the height
of my rhombus then I could subtract the 25 or excuse me the height of the rhombus from
25 and know that there so who remembers what the area of a rhombus is or how to calculate
the area of a rhombus anyone anyone oh yes it's the base 25 times the height. So I know the area
of the rhombus is 500 I know that the base is 25 and so what times 25 gives me 500 it is 20.
so now I know that the height here is 20 so that means that this is going to be 5. Great so now I
know that the area of the rectangle is 25 times 5 times 5 which is equal to 125 excellent so now all
I need to do is find the area of this triangle and then add it to 125 and I'll have my answer.
Um so area of a triangle is one half base times height so for this triangle here um I have the
let's see what do I have here oh I know that this is 20 because this is 5 and a
rhombus is like a square it has all equal sides so this is actually going to be 25
and then I can use the pythagorean theorem to get this side here. So let's do a squared plus
b squared equals c squared so we have 20 squared plus b squared equals 25 squared
this is 400 plus b squared equals 625 b squared. Whoa that's really ugly too, looks like a five
b squared is equal to 625 minus 400 which is 225 the square root of 225 is
81 15. all right so now I know that this side is 15. Put it up here draw an arrow
all right and now I can solve for the area of the triangle so one half base is going to be 20
times 15. So one half of 20 is 10. 10 times 15 is 150 all right are we done no last part we
gotta add these two together 125 and 150 which gives me 275 and my final answer then is 275.
All right excellent I hope that you understand this problem a little bit better now. Um a lot of
times on the GMAT they give you - you know you have to know a lot about geometry and be able to work
at problems in different directions and this is pretty normal where you're working with different
concepts you're combining them and having to do multiple steps to get to your final answer.
So have a wonderful day if you guys have any questions or comments or suggestions
please put them down in the comments, we'll be happy to answer them!
Be excellent to the universe and I will see you next week.