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Might have to stop doing POTW if there is little or no interest in the math extension questions I put up. I will continue if there are even a few people still posting.
POTW The answer to this question is very straightforward. Firstly, I have to find the side lengths of triangle ABC. The possible combinations for side lengths of triangle ABC are listed below. AB= 3,4,12,1,6,2 3x4=12 . 12x1=12 . 6x2=12 BC=3,4,12,1,6,2 4x3=12 1x12=12 . 2x6=12 AC=Not needed After this, I have to multiply the side lengths by 3 and multiply AB by BC, and then I will have to divide by 2. If all of the areas are the same, that is the total area of DEF. 3x3=9 . 3x4=12 . 12x9= 108 . 108/2=54 units squared. 2x3=6 . 6x3=18 . 18x6= 108 . 108/2=54 units squared. 12x3=36 1x3=3 . 36x3=108 . 108/2=54 units squared.
Since all of the areas of DEF with each side length possibilities are the same, 54 units squared is the total area of DEF.
(please continue putting up POTWs. I don’t know about other students, but I like them every week :)
The first thing I decided to do was draw down the two figures and use x, y, z for the side lengths AB, BC, and AC respectively (wow, big gorl words). We know that the area of a triangle is (b x h)/2, and we know that that would equal 6. So… (y*x/2) = 6 y*x = 12 Now, we know that the base and height of the triangle multiply to give us 12. Possibilities must be factors of 12; (1,12); (2,6) and (3,4) and their reciprocals. And since DEF is a similar triangle to ABC (literally), we know that the base and height are 3 times more. We can form the following equation:
(3b x 3h)/2 = area (9bh)/2) = area (magic time: since we know that b x h is 12, we can….) (9x12)/2 = area 9 x 6 = area Area = 54 cm^2.
I noticed that from my equation, there was a 3^2, and after some mathematical sleuthing, this would have been the case for all the side lengths of triangle DEF because it is similar to the side lengths of ABC. Oh, and at the end, when I multipled by 6, that was the area of triangle ABC. Hmmmmmmmmmmmmmmmmmmmmmmmm.
Anyways, therefore, the area of triangle DEF is 54 cm^2.
Given the picture for reference, we can assume that line AB and line BC are the same in length. Using the fact that the area is 6cm^2 we can assume that the triangle is half of a square. If this is true that the area of that square would be two times that of the triangle. If we find the square root of 12 we get 3.464cm. We multiply that by 3 and we get 10.392 cm for one side of the second triangle. 10.392^2 = 108cm^2 (rounded up). But don’t forget that this is a square with 3x the dimensions of the first triangle. We divide that area of the square and make a triangle with the 3x the dimensions of the first. The area of this second triangle is 54 cm^2. (Pls keep the POTWs going, mabye don’t do quadratics tho)
Also, condidering the fact that the area of any triangle is half of height x base 1/2 (h x b) and considering the sides of the second triangle are 1/2 (3h x 3b) = 9 x 1/2(h x b). Therefore the area of the second triangle is 9x the area of the first.
I have been away from POTW for quite a while. Sorry about that. This question was quite straight forward. So first, I turned these triangles into rectangles which are exactly twice its size. I then put 2 more of the same rectangle on top and the the right of the original to simulate the width and length being 3 times as big. Now I am left with a weird L shape. Then, I filled the top right corner with 4 rectangles of the same shape so it looks similar to the original. This rectangle is thrice as wide and long as the original, and it is 9 times as big as the original, so I can assume that the same is the case with triangles (I double checked by squaring 3, which is 9). Since the original triangle is 6 cm squared, I multiply it by 9 to get 54 cm squared, which is the area of the triangle DEF. (Plz continue POTW i tried to get a few people in our class) (We can help with making the questions if you'd like)
Great feedback Tin. Thanks. Get people onboard! And I do have a lot of questions made but I am always open to having you students post some each week too. Just let me know!
Grade 7/8 POTW This question is pretty simple and straightforwards and simple. We know the area of a triangle is (base x height)/2. Since it is a right triangle, the height is a side length and the base is as well. We know the base is 3 times greater and so is the height. Since we are multiplying, 3 x 3 = 9. 9 x 6, the original area, is 54cm^2. Therefore the area of triangle DEF is 54cm^2.
Forgot to do POTW for a couple of weeks. (Keep posting the questions though, I enjoy them)
If the area of this right triangle is 6u^2, the height and length have to multiply to 12, and because the proportion of the sides does not matter (they always multiply to 12), I will make the triangle have an equal base and height. This means the base and height are the square root of twelve. To multiply this number by 3, we actually have to multiple sqrt. 12 and sqrt. 9, since 9 is 3^2. 12*9 is 108, and 108/2 is 54.
Let's say that side AB is 3 cm, and BC is 4 cm. Area of a triangle is (l x w)/2 = A. So, we multiply 3 by 4, giving us 12, and dividing by 2, giving us 6 cm, the area of triangle ABC.
Now for the triangle DEF. I know that line DE is 3AB, therefore line DE would be 3 x 3 which equals 9 cm. Line EF is 3BC, so it would be 3 x 4, which equals 12 cm. Now to find the area of triangle DEF. 12 cm x 9 cm = 108 cm, 108 / 2 = 54 cm squared.
POTW
ReplyDeleteThe answer to this question is very straightforward. Firstly, I have to find the side lengths of triangle ABC. The possible combinations for side lengths of triangle ABC are listed below.
AB= 3,4,12,1,6,2 3x4=12 . 12x1=12 . 6x2=12
BC=3,4,12,1,6,2 4x3=12 1x12=12 . 2x6=12
AC=Not needed
After this, I have to multiply the side lengths by 3 and multiply AB by BC, and then I will have to divide by 2. If all of the areas are the same, that is the total area of DEF.
3x3=9 . 3x4=12 . 12x9= 108 . 108/2=54 units squared.
2x3=6 . 6x3=18 . 18x6= 108 . 108/2=54 units squared.
12x3=36 1x3=3 . 36x3=108 . 108/2=54 units squared.
Since all of the areas of DEF with each side length possibilities are the same, 54 units squared is the total area of DEF.
POTW:
ReplyDelete(please continue putting up POTWs. I don’t know about other students, but I like them every week :)
The first thing I decided to do was draw down the two figures and use x, y, z for the side lengths AB, BC, and AC respectively (wow, big gorl words). We know that the area of a triangle is (b x h)/2, and we know that that would equal 6. So…
(y*x/2) = 6
y*x = 12
Now, we know that the base and height of the triangle multiply to give us 12. Possibilities must be factors of 12; (1,12); (2,6) and (3,4) and their reciprocals. And since DEF is a similar triangle to ABC (literally), we know that the base and height are 3 times more. We can form the following equation:
(3b x 3h)/2 = area
(9bh)/2) = area
(magic time: since we know that b x h is 12, we can….)
(9x12)/2 = area
9 x 6 = area
Area = 54 cm^2.
I noticed that from my equation, there was a 3^2, and after some mathematical sleuthing, this would have been the case for all the side lengths of triangle DEF because it is similar to the side lengths of ABC. Oh, and at the end, when I multipled by 6, that was the area of triangle ABC. Hmmmmmmmmmmmmmmmmmmmmmmmm.
Anyways, therefore, the area of triangle DEF is 54 cm^2.
Thank you for the comment Fiona. Keep going on these and so will I (...in posting them).
DeleteGiven the picture for reference, we can assume that line AB and line BC are the same in length. Using the fact that the area is 6cm^2 we can assume that the triangle is half of a square. If this is true that the area of that square would be two times that of the triangle. If we find the square root of 12 we get 3.464cm. We multiply that by 3 and we get 10.392 cm for one side of the second triangle. 10.392^2 = 108cm^2 (rounded up). But don’t forget that this is a square with 3x the dimensions of the first triangle. We divide that area of the square and make a triangle with the 3x the dimensions of the first. The area of this second triangle is 54 cm^2. (Pls keep the POTWs going, mabye don’t do quadratics tho)
ReplyDeleteThanks for the positive vibes Can!
Deletepsshh, quadratics? totally didn't bring that up last time... pshhh, what are you talking about...
DeleteAlso, condidering the fact that the area of any triangle is half of height x base 1/2 (h x b) and considering the sides of the second triangle are 1/2 (3h x 3b) = 9 x 1/2(h x b). Therefore the area of the second triangle is 9x the area of the first.
ReplyDelete9 x 6 = 56
I have been away from POTW for quite a while. Sorry about that.
ReplyDeleteThis question was quite straight forward.
So first, I turned these triangles into rectangles which are exactly twice its size. I then put 2 more of the same rectangle on top and the the right of the original to simulate the width and length being 3 times as big. Now I am left with a weird L shape. Then, I filled the top right corner with 4 rectangles of the same shape so it looks similar to the original. This rectangle is thrice as wide and long as the original, and it is 9 times as big as the original, so I can assume that the same is the case with triangles (I double checked by squaring 3, which is 9). Since the original triangle is 6 cm squared, I multiply it by 9 to get 54 cm squared, which is the area of the triangle DEF.
(Plz continue POTW i tried to get a few people in our class)
(We can help with making the questions if you'd like)
Great feedback Tin. Thanks. Get people onboard! And I do have a lot of questions made but I am always open to having you students post some each week too. Just let me know!
DeleteGrade 7/8 POTW
ReplyDeleteThis question is pretty simple and straightforwards and simple. We know the area of a triangle is (base x height)/2. Since it is a right triangle, the height is a side length and the base is as well. We know the base is 3 times greater and so is the height. Since we are multiplying, 3 x 3 = 9. 9 x 6, the original area, is 54cm^2.
Therefore the area of triangle DEF is 54cm^2.
typo: *simple and straightforward
DeleteForgot to do POTW for a couple of weeks. (Keep posting the questions though, I enjoy them)
ReplyDeleteIf the area of this right triangle is 6u^2, the height and length have to multiply to 12, and because the proportion of the sides does not matter (they always multiply to 12), I will make the triangle have an equal base and height. This means the base and height are the square root of twelve. To multiply this number by 3, we actually have to multiple sqrt. 12 and sqrt. 9, since 9 is 3^2. 12*9 is 108, and 108/2 is 54.
Therefore, the area of triangle DEF is 54u^2.
Thanks Alan
DeletePOTW:
ReplyDeleteLet's say that side AB is 3 cm, and BC is 4 cm. Area of a triangle is (l x w)/2 = A. So, we multiply 3 by 4, giving us 12, and dividing by 2, giving us 6 cm, the area of triangle ABC.
Now for the triangle DEF. I know that line DE is 3AB, therefore line DE would be 3 x 3 which equals 9 cm. Line EF is 3BC, so it would be 3 x 4, which equals 12 cm. Now to find the area of triangle DEF. 12 cm x 9 cm = 108 cm, 108 / 2 = 54 cm squared.