How to find the area of this triangle

[@bprpmathbasics]

keeping the people maths ready (GCSE geometry problem, Reddit r/GCSE) kzbin.info/www/bejne/a6qYppqaYr-geqs

[@leahithink]

bazinga

[@Itstoearly]

I completed the 2 sides of the rectangle using x and y, then wrote out the area of the 3 white triangles, and using the 2 formulas: Area of shaded = 96 - (area of 3 white triangles), and (y+8)*(x+5)=96, all the variables cancelled out. Not as elegant as your solution, but it worked!

[@SigmaBallistics]

would love to see a video on this

[@L_Ratio_01]

What I did is assigned the horizontal unknown distance be x and vertical unknown distance be y. One equation we can get is that (8+x)(5+y) =96. Now I have the formula memorised for area of triangle through coordinates of vertices (x,0) ,(0,y) ,(8+x , 5+y), which is basically a determinant to solve, and on substituting y, it just beautifully cancels out the x term and gives the area as 28.

[@krischan67]

There's a simpler solution: If that's enough information, then any rectangle with area 96 and the horizontal lenght being at least 8 and the vertical at least 5 will do. So I will choose a rectangle with the horizontal length 8, so the vertical length is 12. That leads to a triangle with a base of 12-5=7 and a height of 8, having an area of 7*8/2=28.

[@mr.e..]

I used a similar method by choosing my rectangle sides, then working out the white triangles and took that away from 96. Example: Vertical side 8, horizontal 12 White triangles are: (5×12)÷2=30 (3×4)÷2=6 (8×8)÷2=32 Total=68 Red triangle=96-68=28 Your example: Vertical 12, horizontal 8 (5×8)÷2=20 (7×0)÷2=0 (12×8)÷2=48 Total=68 Example 3: Vertical 6, horizontal 16 (5×16)÷2=40 (1×8)÷2=4 (6×8)÷2=24 Total=68

[@MinhTran-fl7qg]

An algebraic solution to this is: Let the two sides of the rectangle be 5+x and 8+y Given that the product of two sides is 96, 8x+5y+xy=56 (i) As the rectangle can be divided into four smaller triangles, including the one with the area A we are looking for, A+8(5+x)/2+5(8+y)/2+xy/2=96, or 2A+80+(8x+5y+xy)=192 (ii) From equations (i) and (ii), A=(192-80-56)/2=28

[@SkinnerRobot]

Brilliant!

[@dlevi67]

No idea what your first solution was like, however, if we call a the shorter side of the rectangle, b the longer side, and x the area of the shaded triangle, we know that: ab = 96 but also, since the rectangle is divided into 4 triangles (one of base b, height 5 at the top, another one of base b-8 and height a-5 bottom left, a third one of base 8 and height a bottom right, and the shaded one in the middle), we also have: 5b/2 + (b-8)(a-5)/2 + 8a/2 + x = 96 multiplying out 5/2b + ab/2 +40/2 -5/2b -8a/2 + 8a/2 + x = 96 the terms in 'a' and 'b' cancel out ab/2 + 40/2 + x = 96 but we know ab = 96 48 + 20 + x = 96 from which x = 28 It's possibly slightly longer, but it doesn't need any auxiliary construction.

[@Mnaughten601]

I first found the prime factorization of 96, then found the combinations that had both values that were bigger than 8 and 5, (small list to make). So I knew that the sides must be 8x12 or 6x16. Then I found the area of unshaded triangles of initial picture. Then I subtracted those areas , 96-6-32-30=28 and 96-24-4-40=28 Both rectangle sizes got me to the same answer. Edit: this solution only takes into account that the sides are whole numbers, so unless the problem says with only whole numbers, I don’t think it is absolutely valid.

[@pauldorman]

I too felt the temptation to guess the unknown sides was a bit of a trap, but I got the correct answer from guessing that way, but now that I've thought about it, I understand what he did. I'll add my description of the solution to the main comment thread. The solution doesn't require any assumptions about the missing sides.

[@ausaramun]

Let 96 = (8+x)(5+y) = 40 + 5x + 8y + xy Therefore 56 = 5x + 8y + xy Let 96 = (5)(8+x)/2 + (8)(5+y)/2 + xy/2 + A Therefore 96 = 20 + 2.5x + 20 + 4y + 0.5xy + A Therefore 56 = 2.5x + 4y + 0.5xy Let 56 = 56 5x + 8y + xy = 0.5(5x+8y+xy) + A A = 0.5(5x+8y+xy) = 0.5(56) = 28 QED

[@JJ_TheGreat]

0:31 Area of shaded triangle is 28 - based on finding areas of the other (non-shaded) triangles, adding them up and subtracting from 96. 96-68 is 28… And the dimensions for 96 have to be a width of 12 and a height of 8, so you can subtract to finish the other sizes of the rectangle, they have to add up to those sizes!

[@pauldorman]

It took me a little while to understand, so for others still scratching your heads, read on. Pause the video at the 2 minute mark so you can see all the shaded triangles. First thing to observe is that the larger rectangle is divided in two, and obviously the area of the top rectangle plus the area of the bottom rectangle equals 96. Our excellent teacher has shown us that we can construct an additional triangle which allows us to "fill out" the top and bottom rectangles so that they each contain a single triangle with he same base and height, and therefore half the respective area. The area of all the the shaded triangles together is therefore half the area of the large rectangle, or 48, and all we need to do is subtract the area of the red triangle that spans the top and bottom rectangles, which is simply half its base (5) times height (8): 48 - 20 = 28. QED! I haven't been a student for over 30 years, but I love coming to channels like this to learn different ways of solving problems that I really couldn't appreciate when I was a kid. This was an excellent video. I love a bit of mystery!

[@harishhimanshu8732]

My method was like a Hit and Trial take the area 96 and make its factors 32 * 3 means it will have atleast a 3 in it and then i got that Length would be 8+4 = 12 and Breadth is 8 and I got the all traingle area value as 6 , 30 , 32 and 96 - 68 = 28 I solved it in my mind that when I check the Ans I was so Happy

[@pranavsharma-pw3zv]

Magnificent

[@harishhimanshu8732]

@@pranavsharma-pw3zv thanks

[@Mnaughten601]

The only issue I have is you have another rectangle it could be, 6x16. It also comes out to be the same answer, but without it I’m not sure you could claim to know your answer was the only answer. Edit: I just realized mine still only accounts for whole number solutions, there should be infinite non integer sides to the rectangle, (I assume all of them will bring a final answer of 28)

[@harishhimanshu8732]

@@Mnaughten601 yes you are correct that is also a solution for but the figure here made it is easy to make 8+4 = 12 But this method is just a hit and trial

[@Mnaughten601]

@@harishhimanshu8732 sure, not denying this way is a solution, but without noting the second possible rectangle, most math teachers I have had would probably only give half credit, unless there is some other proof that says if one solution is valid then it is equivalent to all other solutions.

[@fizisistguy]

I couldn't understand. Can you 0lease break down the steps and explain why you did them? (eg - How could you just create the division saying that it is at the half of the length 5+y)

[@balorcamulus5552]

Thank you so much for your lessons.

[@professorvatcraft]

I solved this using algebra :] 1. Give the right side of the rectangle length x -> The left side also has length x -> Top and bottom sides have length 96/x 2. On the left side, we know part of it is length 5 -> The other part of it will be x - 5 3. Do the same for the bottom side -> the other part of it will be 96/x - 8 4. The area of the marked triangle will be the area of the rectangle - the sum area of the 3 small unmarked triangles -> A(MT) = A(R) - A(T1) - A(T2) - A(T3) (*) A: Area MT: Marked triangle R: Rectangle T1: Triangle 1 (has base x and height 8) T2: Triangle 2 (has base 5 and height 96/x) T3: Triangle 3 (has base x - 5 and height 96/x - 8) 5. Area of triangle 1: A(T1) = 1/2 * x * 8 = 4x 6. Area of triangle 2: A(T2) = 1/2 * 96/x * 5 = 240/x 7. Area of triangle 3: A(T3) = 1/2 * (x-5)(96/x - 8) = 48 - 4x - 240/x + 20 = -4x - 240/x + 68 8. Filling the values in to (*) A(MT) = 96 - 4x - 240/x - (-4x - 240/x + 68) A(MT) = 96 - 4x - 240/x + 4x + 240/x - 68 A(MT) = 96 - 68 A(MT) = 28 ✅ I assume this is how most of us solve this problem :] Your solution is really interesting! I did not think you can do that :D

[@adipy8912]

Finally more geometry

[@raymundbelleza1279]

I did this through measuring and got 28.6.

[@zachansen8293]

it's funny this one I find to be much simpler using algebra and can't follow your explanation at all. Just find the area of the other three triangles in terms of X and Y, then substitute either x for y or y for x using the x*y=96 (x=96/y) or (y=96/x) given and then subtract from the total area.

[@Anmol_Sinha]

THIS IS GENIUS!!

[@Nothingx303]

😊 ευχαριστώ για αυτό ερώτησή

[@turmericgarage8509]

For the first time I paused when asked and tried to do it in my head. Then got lazy and guessed if 96 is 12x8, then the bottom left white is a 3 4 5 of area 6. The other two white triangles are area 30 and 32. Thus 28 for the shaded area.

[@tawney709]

I don’t understand how this works. If the sides of the rectangle were more than 10 wouldn’t that throw the whole thing off? Was it said that 5 was half of the length? Isn’t this also just assuming the shaded triangles equal half of the rectangle?

[@RAG981]

Neat way.

[@Gremriel]

I swear you did this one before, and I was about to ask AndyMath what he thought about it, but then the video got deleted.

[@bprpmathbasics]

I did! And I didn’t like what I did. So I private that video.

[@shreyas7346]

how do we know the area of the bottom red and black triangle is equal to half the area of the bottom rectangle?

[@pranavsharma-pw3zv]

Pick a random Rectangle: (Say sides) A,B Pick any one vertex of ur choice and then measure a length L on on side such that L is less than the side u picked Now do some basic geometry and try to calculate the area of the triangle created U will end up with Area=(AB/2) always (The result does not depend on L)

[@borchen0]

He forgot to draw a straight line up, beginning at the point where the original triangle meets the bottom line...

[@russelltownsend6105]

Make a vertical cut in in the lower rectangle from the lower triangle vertex to meet the first horizontal cut this divides the lower rectangle into 2 smaller rectangles. These smaller rectangles are halved by the diagonals of each of these smaller rectangles that make the boundary of the lower part of that shape. This proves that the portion of the shades area below the horizontal cut construct is half of the area of the lower large rectangle.

[@robertpearce8394]

​@russelltownsend6105 Yes. It puzzled me for a bit. It could have been made clearer.

[@NavalSharma-r9z]

nice video

[@pranavsharma-pw3zv]

3:04 I have a strong feeling that u did make a blunder right at this part Ig u subtracted more than what was required

[@nekoDentaku]

wdym? it's correct

[@pranavsharma-pw3zv]

@@nekoDentaku is the answers right, But Ig i still dont get a point maybe