Calculate the area of all rectangles | Area of Blue rectangle is 56

Calculate the area of all rectangles | Area of Blue rectangle is 56 | Important skills explained

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Learn how to find the area of all rectangles by solving for X. Important Geometry and algebra skills are also explained: area of the rectangle formula. Step-by-step tutorial by PreMath.com
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Calculate the area of all rectangles | Area of Blue rectangle is 56 | Important skills explained
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Пікірлер: 57

@jeminginjemingin780 Жыл бұрын

Thanks sir to teaching me out the box💪

@PreMath Жыл бұрын

Happy to help Thanks for your feedback! Cheers! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

@j.r.1210 Жыл бұрын

The interesting thing about your solution is that the key step is multiplying a x b x c x d -- something that has no *geometric* meaning here, but is crucial for generating a solvable equation. Neat.

@e1woqf Жыл бұрын

No, it's not crucial. Indeed there is no need to introduce a,b,c and d. Since both the upper and the lower rectangles share one side respectively, we can work with the ratio of areas, which leads us to the equation 56 (x+2) = (x-4)(4x-3)

@murdock5537 Жыл бұрын

@@e1woqf = (x - 4)(4x - 3) 🙂

@pwmiles56 Жыл бұрын

Nice problem. You can do it by ratios of the areas green/purple = brown/blue (x+2)/(4x - 3) = (x - 4)/56 Multiply out denominators and simplify 4 x^2 - 75x -100 = 0 x = (75 +/- sqrt(75^2 + 1600))/8 x = (75 +/- 5 sqrt(15^2 + 64))/8 x = (75 +/- 5 sqrt(289))/8 x = (75 +/- 85) / 8 x = -5/4 or 20 etc

@e1woqf Жыл бұрын

Exactly.

@geraldillo Жыл бұрын

It's the same; Premath skipped that step (unfortunately)

@miguelgnievesl6882 Жыл бұрын

If I multiply the diagonal areas and then equalize those products [56(x+2)] = [(4x-3)(x-4)], I get the same quadratic equation and of course the same values for x.

@e1woqf Жыл бұрын

That's exactly what I did as well. No need to introduce extra variables.

@alster724 Жыл бұрын

The ratio-and-proportion technique

@krzysz5023 Жыл бұрын

Yeah I noticed that the radio of the area of the boxes would be proportional to the ratio of the corresponding sides (after doing a bunch of other work) and got that too! 😅

@santiagoarosam430 Жыл бұрын

The ratio between areas of the same column or row is constant throughout the array of rectangular cells → (X+2)/(4X-3=(X-4)/56 → X=20 → X+2=22 ; X -4=16 ;4X-3=77 If the dimensions of the purple cell are B*H and we put the dimensions of the other column and the other row as a function of B and H, we obtain a column width =B*b and a row height =H*h → Result some areas: Green=BHh ; Brown=BbHh ; Purple=BH ; Blue=BbH → With these algebraic expressions we can check the constancy of the relationships between areas of the same column or row: Green/Purple=BHh/BH=h = Brown/Blue=BbHh/BbH=h ; Green/Brown=BHh/BbHh=1/b = Purple/Blue=BH/BbH=1/b. Thanks and greetings to all.

@mfrankin6554 Жыл бұрын

That was a challenging problem. I had to really think through the logic and then it occurred to me that essentially you solved it as a proportion (x+2)/(4x-3) x (x-4)/56. Only that you did it by labeling the sides and then cross multiplying. That was was excellent. Your problems really get me thinking outside of the box.

@alster724 Жыл бұрын

The manipulation part of abcd was a bit tricky but the rest are easy. The ratio-proportion technique of the labelled areas would've been easier AB/AC=AD/CD (x+2)/(4x-3)=(x-4)/56

@vaibhavgupta5359 Жыл бұрын

Your channel is a treasure of knowledge

@timeonly1401 Жыл бұрын

For 2 rectangles with a common length (A = l*w, & B = l*y), the ratio of their areas is equal to the ratio of their widths (A/B = lw/ly = w/y). We see that the 2 rectangles in the 1st column have the same lengths (as labelled in diagram: b), so the ratio of their areas is a/c. Likewise for the rectangles in the 2nd column, the ratios of their areas is also a/c. Therefore the two ratios of areas are equal to each other. Therefore, (x+2)/(4x-3) = (x-4)/56. Then solve for x, then get the areas of the 3 unknown areas. ^^Notice: As a mnenomic, you just set up the ratios EXACTLY in the same position as the subrectangles are positioned in the big, overall rectangle!! If you remember this, you can just write the equal ratios, and off you go!! (If you aren't sure how the ratios go, you can always derive this from scratch, which should only take 30-60 secs.)

@MrPaulc222 Жыл бұрын

I went with 56(x+2) = (4x-3)(x-4) , formed the quadratic from that, and calculated x with quadratic formula.

@nineko Жыл бұрын

Another nice one, the best part is that you can also calculate the individual lengths of the small segments: 2 and 7 on the vertical axis, 11 and 8 on the horizontal axis, and it all works out. In fact, 7 • 8 was my first guess for the blue rectangle even before you started your procedure, because 56 doesn't have many other suitable divisors.

@owlsmath Жыл бұрын

Thanks! Great problem - like the way it mixes geometry with the algebra

@murdock5537 Жыл бұрын

Nice problem, excellent way to solve it, many thanks! fast lane: (x + 2)/(4x - 3) = (x - 4)/56 → (x + 2)56 = (4x - 3)(x - 4) → x = 20 btw: a = 2; b = 11; c = 7 ; d = 8 🙂 56 = 7(8) → 77 = 7(11) → 22 = 2(11) → 16 = 2(8)

@theoyanto Жыл бұрын

Yeah, these are the ones that cause me trouble in getting started.... So more like this please 🤓👍🏻

@HappyFamilyOnline Жыл бұрын

Amazing👍 Thanks for sharing😊

@KAvi_YA666 Жыл бұрын

Thanks for video.Good luck sir!!!!!!!!!!

@mehulpunia6174 Жыл бұрын

Very interesting sir

@nirupamasingh2948 Жыл бұрын

Vvv nice explanation

@alexrocha9191 Жыл бұрын

Good Morning Master 🇧🇷

@johnbrennan3372 Жыл бұрын

Divide (1)by (2) and (3) by (4) each of which =b\d. Then cross multiply etc.

@e1woqf Жыл бұрын

👍

@johnbrennan3372 Жыл бұрын

Thank you

@bigm383 Жыл бұрын

That was a fun problem. Thanks professor!😄

@PreMath Жыл бұрын

Glad to hear that! Thanks for your feedback! Cheers! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

@yalchingedikgedik8007 Жыл бұрын

Thats good idea

@williamspostoronnim9845 Жыл бұрын

Задачка простая, пропорция составляется сразу же из подобия прямоугольников.

@vladtepes97 Жыл бұрын

4:15 step 3, why did you multiply the area of rectangle ab by the area of rectangle cd?

@aryabhattagamharia5586 Жыл бұрын

Amazing sir

@PreMath Жыл бұрын

Many many thanks You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

@Tomaslyftning 9 ай бұрын

We dont need to calculate the sides if we notice that blue to brown = purple to green: 56/(x-4) = (4x-3)/(x+2) then cross multiplication and simplification give: 4xx - 75x - 100 =0 and the positive solution X=20

@clapptschon Жыл бұрын

I got the Idea: 56 / (x-4) = (4x-3) / (x+2) => x = 20 kind regards

@PreMath Жыл бұрын

Excellent! Glad you think so! You are awesome, Michael. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

@devondevon4366 Жыл бұрын

16 , 22, 77, 56 (given) and 171

@mahalakshmiganapathy6455 Жыл бұрын

Super

@PreMath Жыл бұрын

Thank you! Cheers! You are awesome, Mahalakshmi. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

@devondevon4366 Жыл бұрын

Answer =16 , 22, 56, 77 and total area =171 or 19 x 9 A different approach. let the blue rectangle area = np (n= the horizontal line and p= the vertical line) let the vertical line of the beige rectangle= r; hence its area = nr let the horizontal line of the green rectangle = y; hence its area = ry let the vertical line of the pink rectangle = p ; hence its area= py Since np =56 and nr = x-4, then p/r = 56/(x-4) [ divide np by nr] Since py= 4x-3 and ry =x+2, then p/r = (4x-3)/x+2) [divide py by ry] Therefore 56/(x-4) = (4x-3)/(x+2) since both = p/r 56x + 112 = 4x^2 -19x +12 [ Crossmultiply] 0 = 4x^2 -75x -100 [ Simplify] quadratic equation. Using the quadratic formulae x =20 hence area of each is 16 (20-4) 22 ( 20 + 2) 77 (4 x 20 -3) 56 (given) Area of the large triangle = 171 or 19 x 9

@pavel-gubanov Жыл бұрын

S(orange)/S(blue) = S(green)/S(purple) => 56(x+2)=(4x-3)*(x-4) => x=20

@richardslater677 Жыл бұрын

Why does abcd = (ab)(cd) etc?

@sandanadurair5862 Жыл бұрын

Real lateral thinking. I was nowhere near this trick.

@awandrew11 Жыл бұрын

What is the meaning of abcd?

@e1woqf Жыл бұрын

a times b times c times d

@pranavamali05 Жыл бұрын

Thnku

@PreMath Жыл бұрын

Thank you! Cheers! You are awesome, Pranav. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

@elianagotilla6564 Жыл бұрын

I didn't get Step 3 at all. Are you multiplying the area of 2 rectangles ? (ab)*(cd) ? Why ? Thank you.

@kurtecaranum3047 Жыл бұрын

It's because Commutative Property comes into play to help us solve x easier (Green)(Blue) = (Orange)(Pink) (ab)(cd) = (ad)(bc) = abcd (x+2)(56) = (x-4)(4x-3) = abcd

@JPTaquari Жыл бұрын

Nice, X + 2 - X - 4 4X-3 - 56 56X + 112 = 4X² - 16X - 3X + 12 4X² - 75X - 100 = zero X = 20 Total area = 171 Bingo !!!!!!!!!

@giuseppemalaguti435 Жыл бұрын

A=6x+51...trovo dai dati x=20,per cui A=171

@Waldlaeufer70 Жыл бұрын

I found a solution via the area ratios of 2 neighbouring rectangles respectively. The variables a, b, c and d are not required in this case: (x + 2) : (4x - 3) = (x - 4) : 56 56 (x + 2) = (x - 4) (4x - 3) 56x + 112 = 4x² - 3x - 16x + 12 4x² - 75x - 100 = 0 x = 20 (x = -1.25 => negative areas!) A = (x + 2) + (x - 4) + (4x - 3) + 56 A = (20 + 2) + (20 - 4) + (80 - 3) + 56 A = 22 + 16 + 77 + 56 = 171 Or if just the complete area is asked: A = 6x + 51 = 6 * 20 + 51 = 120 + 51 = 171