What is the blue area? (remastered)

#math #geometry

This is a quick mathematics/geometry puzzle. Elementary geometric properties are required to solve this problem. Given two squares, compute the area of the blue 5-gon shape.

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Keywords: geometry, problem solving, math problems, math puzzle, trigonometry, pythagoras, angle chasing, animation.

[ictmathematics]

Sorry for the small-sized texts, next videos will have larger texts.

[boanergesct]

L= Blue square side; l=Side red square → Area red square=1+3=4 → l=√4=2 → Area red outer triangle= 1=2a/2 → a=1 → Hypotenuse=b → 2²+1²=b² → b= √5 → The red and green triangles are similar → Hypotenuse of the green triangle = a=1 → Similarity ratio between both triangles =1/√5= √5/5 → Green triangle sides: (√5/5), (2 √5/5), (1) → L= (√5)+(√5/5)= (6√5)/5 → Green triangle area= (1/2) (√5/5) (2√ 5/5)= 1/5=0.20 → Blue irregular pentagon area= L²-(Green triangle)-(Inner red area)= (6√5/5)²-(1/5)-3= 20/5= 4
Another solution: Once the hypotenuse of the outer red triangle is found, it is immediate to see that the large square can be divided into 3x6=18 rectangular cells with the dimensions of the green triangle; of those cells 10 correspond to the blue area → 10x2/5=4
Ultreya!! friends of the road

[geoninja]

I thought your reasoning behind showing the green and red triangles are similar was a bit complex - they share right angles and another vertical angle pair, so are clearly similar, without needing to prove rotations.... great video though....

[gelbkehlchen]

Solution:
The area of the red square is 1+3=4, so one side of the red square is √4=2.

The red triangle has the area 1, one leg is 2, the other leg is a.
Then: 2*a/2=1 ⟹ a=1 ⟹ The hypotenuse of the green triangle is 2-1=1.

The hypotenuse of the red triangle is: √(2²+1²)=√5.

From the similarity of the green triangle with the red triangle follows:
short leg from the green triangle / hypotenuse from the green triangle
= short leg of the red triangle / hypotenuse of the red triangle ⟹
short leg of the green triangle = 1*1/√5 = 1/√5

One side of the big square is then √5+1/√5=(5+1)/√5=6/√5 and the area of the big square is then (6/√5)²=36/5=7, 2.

From the similarity of the green triangle with the red triangle follows:
long leg of the green triangle / hypotenuse of the green triangle
= long leg of the red triangle / hypotenuse of the red triangle ⟹
long leg of the green triangle = 1*2/√5 = 2/√5

The area of the green triangle is then 1/2*1/√5*2/√5=1/5=0, 2.

The blue pentagon area is now: 7, 2-3-0, 2=4

[Nikioko]

Since the red triangle is 1/4 of the red square, the short leg (length 1) must be half the long leg (length 2). We could now calculate that the hypotenuse is √5 and go on, but there is an easier way:
Draw a perpendicular to the upper and lower side of the blue square through the right corner of the red square. Left of this line, you get two right triangles, which are identical to the red triangle. Therefore, the blue square has a side length of 1 + 2 = 3. The difference between the blue and the red square is 3² - 2² = 5. Since the area of the red triangle is outside the blue square, you have to add 1 and get 6. From that, you have to subtract d the green triangle of which you know that its hypotenuse is 1 and its legs have the ratio 1:2:
x² + (2x)² = 1
5x² = 1
x² = 1/5
Which is also the area of the green triangle. Subtract the from the 6 square units, and we get 5, 8 square unit as the result for the blue pentagon.

[subaqua]

Outstanding content!
Here are some tips:
1- Make the video faster
2- Use one music track

Great video nonetheless!

[mikefochtman]

Well I skipped a little bit of Pythagoras. Once noticed that sides of green triangle were 1/sqrt(5) of larger red triangle, it was obvious the area of green was (1/sqrt(5))^2 of red triangle. (areas are related by (scaling factor)^2). Not a big difference in solution.

Love the animations though and yes on the slightly larger text next time.

[JLvatron]

Great math video! Music was a bit creepy.

[JoeGelman]

The numbers here line up weirdly well. Coincidence, or is there some kind of generalizable proof here?

[doubledee]

Not really enough explanation of the intermediate steps.