Find the Area of the Blue Triangle Inside of a Rectangle

In this video we use the formulas for the areas of rectangles and triangles, as well as the quadratic formula to find the area of the blue triangle.

[spacelemur]

As a 70 y.o. man who hasn't worked with these types of problems for >45 years, I could visualize this approach in principle, but couldn't remember the quadratic equation anymore and had to look it up. It also took me much longer than the video lasted.

[crustyoldfart]

When you have 3 equations and 4 unknowns, it's important to recognize that a solution for a*b is sufficient.

[dongxuli]

we have 3 equations: bx=54; ay=30; (b-y)(a-x)=24; add the 3 equations together, we have: ab + xy = 108; multiply the first two equations together, we have (ab) (xy)=54*30. Therefore, both the sum and the product are known for the two numbers ab and xy, and they are the two roots of the quadratic equation: U^2 - 108 U + 54*30 = 0. Solving the quadratic equation, we have ab=90, xy=18 (since ab > xy). Finally, the answer to the original question is ab - 27-15-12 = 36.

[benw]

As an engineer this is what I hate about math, they intentionally give you the minimum amount of information. In reality, you almost always have enough information to solve by multiple means.

[lybcxds]

تمرين جميل جيد . رسم واضح مرتب . شرح واضح مرتب . شكرا جزيلا لكم والله يحفظكم ويرعاكم ويحميكم جميعا. تحياتنا لكم من غزة فلسطين

[devondevon]

Did without pen and paper by assuming the values are integers
Starting with the triangle of area 15, the sides are 3* 10 ( of course, 2*15, 6*5 are possible, but 3*10 looks more reasonable). Hence, one side of the rectangle is 10 for now, and part of the
other side =3
For the triangle with area 12, L*W = 24. Let's try 6* 4, and add 6 + 3 = 9. Hence 4 goes
to the other side, which for now is 10. Hence we need a 6 from the triangle with an area of 2
For the triangle with this area L* W =54. Let's use 6* 9 since we are looking for a 6 and 9 .
Hence the sides for the rectangle are 9 and 10. Hence the area of the rectangle = 90. And the area
of the triangle:
90-(27-15- 12 ) = 90 - 54 = 36 Answer

[BytebroUK]

I watch a fair few vids of this kind of thing, but I must say your explanations are very clear and concise. And yes, there are several ways of doing that, but you chose one and explained it very well.

[jpopelish]

I like how the answer does not constrain the actual values of a and b, individually, but shows an infinite family of triangles in an infinity of rectangles satisfy this set of constraints.

[evo]

It's not a mathematical way of deriving, but if you multiply the area of each triangle by doubling it, that is, assume the length of the side of the quadrangle as the area of the quadrangle, you get the following.
27 = (1/2)*9*6 ... x = 6, b = 9
12 = (1/2)*4*6 ... (a-x)=4, (b-y)=6
15 = (1/2)*3*10 ... y=3, a=10

b = (b-y) + y = 6 + 3 = 9
a = x + (a-x) = 6 + 4 = 10
a * b = 90
Blue Triangle = 90 - 27 - 12 - 15 = 36

[guilhermeviana]

Excelente. Bastante didático. Muito útil.

[tungyeeso]

I like your approach as much as I like your accent. The former is appealing, the latter comforting. Thanks.

[Gwydion_Wolf]

its 330am and this popped up on my recommended for some reason :)

Only 6 seconds in (wanted to make sure the numbers shown were the 'area' of the known triangles). But here's how i'd go about solving this.

1) Solve the A2, B2, C2, for all 3 known triangles.
2) add the two 'lengths' (ie: a2 b2) on the left that connect 27 and 12, and add the lengths of where 12 and 15 meet.
3) use that to figure out the total area of the rectangle formed.
4) subtract the total area of the 3 triangles.
5) left with the area for the blue triangle?

:)

*quick edit* When i say "solve for a2, b2, c2" i also mean to take into account that the two sides that connect 12 and 15 at the bottom combined must also = the x2 that makes up the top side of 27, and that the x2's for 27 and 12 = that of the right side of 15

[Wmann]

Solution is simple to understand, but needs a creative mind to produce. Neatly done.

[escapistdesign]

I like the way you explain everything. Nice video.

[sumantabanerjee]

Reminded me of my good old school days!! Thanks, Maa'm, for the nice explanation!! 🙏😊

[richardleveson]

Really nice explanation - and a nice problem too! Thank you

[geoffreyparfitt]

The important feature of this type of problem is that the blue area is exactly defined but the dimensions of the rectangle are not unique. Therefore one dimension of the rectangle can be conveniently decided by you, such as a width of 9 units, or a height of 10 units. Either of those choices leads to a simple quadratic that gives the other dimension and consequently the area of the rectangle.

[icems.a.]

Muy bueno, como los otros que he visto en este canal . 👍

[shadrana]

Using Pythagoras, I calculated the sides of the blue triangle sqrt(117), sqrt(52) and sqrt(109) and using a Herons Formula calculator and surprise, surprise the area of the blue triangle came out at 36 sq. units.
let a=sqrt117, b=sqrt52 and c=sqrt109
Heron's formula; in terms of the sides a, b and c.


=sqrt(208*117-60^2)/4
=sqrt(24336-3600)/4
=sqrt(20736)/4
=144/4
=36 sq.units.
This checks our answer out using Pythagoras and Heron.

[The_NSeven]

Great solution. Explained it very well :)