I give thanks Jumar for the suggestion!

A square has a side length of l cm. Four right triangles with legs of 40 cm and thirty cm are placed along the four corners of the foursquare, every bit shown below.

What is the area of the region inside the foursquare not covered by the four triangles? In other words, what is the area of the region shaded in light-green?

As usual, watch the video for a solution.

Four Triangles In A Square Puzzle

Or continue reading.
.
.

"All will be well if you use your heed for your decisions, and mind just your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions at present has over 1,000 free articles with no ads cheers to community support! Help out and become early on access to posts with a pledge on Patreon.

.
.

.
.
.
.
M
I
Northward
D
.
Y
O
U
R
.
D
E
C
I
S
I
O
N
S
.
P
U
Z
Z
L
East
.
.
.
.
Answer To Iv Triangles In A Foursquare Puzzle

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will right them, thanks).

Consider 2 adjacent right triangles. Since each has legs of 30 and 40, each has a hypotenuse equal to 50 (each is a 3-4-5 right triangle scaled by 10). The two triangles overlap to form another triangle shown in imperial.

Because the two triangles are rotated 90 degrees, their hypotenuses meet at a right angle. And then the two angles in the purple triangle are one of each of the acute angles of the blueish triangle. Thus the purple triangle is similar to the bluish triangle.

Now we will solve for the hypotenuse of the royal triangle. The small leg of the blue triangle is thirty, and then the distance from angle A to the right corner of the square is fifty – 30 = 20. And then the longer leg of the blueish triangle is 40, so the altitude from bending B to the left corner of the foursquare is 10. This means the hypotenuse of the majestic triangle is 50 – x – 20 = 20.

The blue triangle has an expanse equal to 30×40/2 = 600. The royal triangle is similar to the blue triangle, and their hypotenuses are in a ratio of 20/fifty = two/v. Thus the regal triangle'due south surface area to the blue triangle's is the squared ratio (2/5)2, and then its area is 600(2/five)2 = 96.

We can so observe the dark-green region'south area past adding and subtracting the areas of relevant shapes.

The green region's surface area is equal to the entire square's expanse minus 4 times a blue triangle'south surface area plus 4 times the purple triangle's expanse (nosotros demand to add dorsum the overlapping regions). Thus we accept:

dark-green area
= square's area – 4(bluish triangle) + 4(majestic triangle)
= fifty2 – 4(600) + 4(96)
= 2500 – 2400 + 384
= 484

Thus the light-green region has an surface area equal to 484.

PRESH TALWALKAR

I run the MindYourDecisions channel on YouTube, which has over i million subscribers and 200 million views. I am also the author of The Joy of Game Theory: An Introduction to Strategic Thinking, and several other books which are available on Amazon.

(As you lot might expect, the links for my books go to their listings on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the toll you pay.)

By manner of history, I started the Mind Your Decisions blog back in 2007 to share a bit of math, personal finance, personal thoughts, and game theory. It'south been quite a journeying! I thank anybody that has shared my work, and I am very grateful for coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, and many other pop outlets.

I studied Economics and Mathematics at Stanford University.

People frequently ask how I make the videos. Similar many YouTubers I utilise popular software to ready my videos. Yous tin can search for blitheness software tutorials on YouTube to larn how to make videos. Be prepared--blitheness is time consuming and software can be expensive!

Feel costless to send me an email [email protected]. I get so many emails that I may not respond, but I save all suggestions for puzzles/video topics.

MY BOOKS

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Book ratings are from January 2022.

(United states and worldwide links)
https://mindyourdecisions.com/weblog/my-books

Mind Your Decisions is a compilation of 5 books:

(1) The Joy of Game Theory: An Introduction to Strategic Thinking
(2) 40 Paradoxes in Logic, Probability, and Game Theory
(3) The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias
(4) The Best Mental Math Tricks
(5) Multiply Numbers Past Drawing Lines

The Joy of Game Theory shows how you tin utilize math to out-think your contest. (rated 4.2/v stars on 224 reviews)

twoscore Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. (rated four.one/5 stars on 38 reviews)

The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased nigh decision-making and offers techniques to brand smart decisions. (rated 4/v stars on 24 reviews)

The All-time Mental Math Tricks teaches how you can look like a math genius past solving issues in your head (rated four.2/5 stars on 76 reviews)

Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 meg views on a geometric method to multiply numbers. (rated 4.three/5 stars on 30 reviews)

Mind Your Puzzles is a collection of the three "Math Puzzles" books, volumes 1, two, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.

Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for issues in counting, geometry, probability, and game theory. Book i is rated 4.4/v stars on 87 reviews.

Math Puzzles Volume ii is a sequel book with more than great problems. (rated 4.one/v stars on 24 reviews)

Math Puzzles Volume 3 is the third in the series. (rated iv.two/v stars on 22 reviews)

KINDLE UNLIMITED

Teachers and students effectually the world frequently email me about the books. Since pedagogy can have such a huge impact, I effort to brand the ebooks bachelor as widely as possible at as low a toll as possible.

Currently you tin can read well-nigh of my ebooks through Amazon's "Kindle Unlimited" program. Included in the subscription y'all will become access to millions of ebooks. You don't need a Kindle device: you can install the Kindle app on whatsoever smartphone/tablet/reckoner/etc. I have compiled links to programs in some countries below. Please cheque your local Amazon website for availability and program terms.

United states of america, listing of my books (Usa)
Britain, list of my books (UK)
Canada, book results (CA)
Germany, list of my books (DE)
France, list of my books (FR)
Republic of india, listing of my books (IN)
Australia, book results (AU)
Italian republic, list of my books (It)
Spain, list of my books (ES)
Japan, listing of my books (JP)
Brazil, book results (BR)
United mexican states, volume results (MX)

Trade

Grab a mug, tshirt, and more at the official site for merchandise: Mind Your Decisions at Teespring.