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06 Jan 2022, 03:18
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
45% (02:59) correct
55%
(02:49)
wrong
based on 82
sessions
History
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Not Attempted Yet
If the side lengths of a triangle are measured in centimetres and the measure of each of the side is a distinct prime number, then the area of the triangle (in cm2) with the least perimeter will be between: [Note: √2 = 1.414 and √3 = 1.732]
A. 5 and 6
B. 6 and 7
C. 7 and 8
D. 8 and 9
E. 9 and 10
A. 5 and 6
B. 6 and 7
C. 7 and 8
D. 8 and 9
E. 9 and 10
17 Jan 2022, 16:20
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Take the smallest right triangle area we know: a triangle with sides 3-4-5. The area of this is 1/2 * 3*4=6.
List out the first few prime numbers: 2, 3, 5, 7.
We know each side must be a prime number and we also know the geometry rule that in order to be a legit triangle, any two sides must add up to be greater than the third side! This is the key to the question.
So we try sets of 3 numbers from the prime number list keeping this rule in mind.
2, 3, 5. Is this a legit triangle? No, because 2+3 is not greater than 5.
3, 5, 7. Is this a legit triangle? Yes because 3+5>7, 5+7>3, and 3+7>5. Stick with this scenario. Now I need to find the area of this triangle but I know it's not a right triangle so this may be difficult. It's okay though because the answers are approximations and I know the area must be around 6 since this triangle's base and height are not far off from that of the 3-4-5 triangle we started with. This gets us down to answer choices A and B. Now we need to figure out: is the area of this triangle greater than or less than 6?
We know the base, 3, is the same for the 3-4-5 and 3-5-7 scenarios. Now what about the height? The second side (5) has to be at an obtuse angle to the base in order to allow the third side to be longer at 7, which will make the height less than 5. However, 7 is not incredibly larger than 5 so the height shouldn't decrease too much so that it goes below 4. Thus, the height is probably still a little greater than 4 so the best guess is B.
Let me know any more efficient ways of solving this.
List out the first few prime numbers: 2, 3, 5, 7.
We know each side must be a prime number and we also know the geometry rule that in order to be a legit triangle, any two sides must add up to be greater than the third side! This is the key to the question.
So we try sets of 3 numbers from the prime number list keeping this rule in mind.
2, 3, 5. Is this a legit triangle? No, because 2+3 is not greater than 5.
3, 5, 7. Is this a legit triangle? Yes because 3+5>7, 5+7>3, and 3+7>5. Stick with this scenario. Now I need to find the area of this triangle but I know it's not a right triangle so this may be difficult. It's okay though because the answers are approximations and I know the area must be around 6 since this triangle's base and height are not far off from that of the 3-4-5 triangle we started with. This gets us down to answer choices A and B. Now we need to figure out: is the area of this triangle greater than or less than 6?
We know the base, 3, is the same for the 3-4-5 and 3-5-7 scenarios. Now what about the height? The second side (5) has to be at an obtuse angle to the base in order to allow the third side to be longer at 7, which will make the height less than 5. However, 7 is not incredibly larger than 5 so the height shouldn't decrease too much so that it goes below 4. Thus, the height is probably still a little greater than 4 so the best guess is B.
Let me know any more efficient ways of solving this.
03 Mar 2023, 05:25
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A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.
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A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.
This post was generated automatically.
