With four congruent isosceles right triangles, can I form a square? a rhombus? a rhomboid?

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1
down vote

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If I have four isosceles right triangles congruent to each other, which of the following figures can I create?



  • I) Square

  • II) Rhombus

  • III) Rhomboid



I) Well, a square is always four isosceles triangle rectangles and congruent, so it fits perfectly. Then, this is it.



II) A square, is a special case of rhombus, but with more properties for which every square is a rhombus. Then, this is it.



III) The same argument of II, for the rhomboid.




So, my answer is I), II), III)



But the correct answer is I) and III).




Well, answer I) must be completely correct, but the argument I used in II and III, I thought was correct, but it is not. So,



Why the answer is I) and III)?




geometry




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  • 2




    Because they're not thinking like mathematicians. As you point out, you can't make (I) without making (II) at the same time. Undoubtedly, they mean a non-square rhombus, but then they should've said so.
    – Brian Tung
    Jul 24 at 2:03






  • 1




    The rhomboid they have in mind is probably the one that has 45 and 135 degree angles, with one pair of sides $sqrt2$ times the length of the other pair.
    – Brian Tung
    Jul 24 at 2:05










  • Brian, how you deduce that?
    – Mattiu
    Jul 24 at 22:05










  • Which part? If you mean the rhomboid part, I just visualized the figure shown in Parcly Taxel's answer.
    – Brian Tung
    Jul 24 at 22:34














up vote
1
down vote

favorite













If I have four isosceles right triangles congruent to each other, which of the following figures can I create?



  • I) Square

  • II) Rhombus

  • III) Rhomboid



I) Well, a square is always four isosceles triangle rectangles and congruent, so it fits perfectly. Then, this is it.



II) A square, is a special case of rhombus, but with more properties for which every square is a rhombus. Then, this is it.



III) The same argument of II, for the rhomboid.




So, my answer is I), II), III)



But the correct answer is I) and III).




Well, answer I) must be completely correct, but the argument I used in II and III, I thought was correct, but it is not. So,



Why the answer is I) and III)?




geometry




share|cite|improve this question

















  • 2




    Because they're not thinking like mathematicians. As you point out, you can't make (I) without making (II) at the same time. Undoubtedly, they mean a non-square rhombus, but then they should've said so.
    – Brian Tung
    Jul 24 at 2:03






  • 1




    The rhomboid they have in mind is probably the one that has 45 and 135 degree angles, with one pair of sides $sqrt2$ times the length of the other pair.
    – Brian Tung
    Jul 24 at 2:05










  • Brian, how you deduce that?
    – Mattiu
    Jul 24 at 22:05










  • Which part? If you mean the rhomboid part, I just visualized the figure shown in Parcly Taxel's answer.
    – Brian Tung
    Jul 24 at 22:34












up vote
1
down vote

favorite









up vote
1
down vote

favorite












If I have four isosceles right triangles congruent to each other, which of the following figures can I create?



  • I) Square

  • II) Rhombus

  • III) Rhomboid



I) Well, a square is always four isosceles triangle rectangles and congruent, so it fits perfectly. Then, this is it.



II) A square, is a special case of rhombus, but with more properties for which every square is a rhombus. Then, this is it.



III) The same argument of II, for the rhomboid.




So, my answer is I), II), III)



But the correct answer is I) and III).




Well, answer I) must be completely correct, but the argument I used in II and III, I thought was correct, but it is not. So,



Why the answer is I) and III)?




geometry




share|cite|improve this question














If I have four isosceles right triangles congruent to each other, which of the following figures can I create?



  • I) Square

  • II) Rhombus

  • III) Rhomboid



I) Well, a square is always four isosceles triangle rectangles and congruent, so it fits perfectly. Then, this is it.



II) A square, is a special case of rhombus, but with more properties for which every square is a rhombus. Then, this is it.



III) The same argument of II, for the rhomboid.




So, my answer is I), II), III)



But the correct answer is I) and III).




Well, answer I) must be completely correct, but the argument I used in II and III, I thought was correct, but it is not. So,



Why the answer is I) and III)?





geometry





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share|cite|improve this question




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edited Jul 24 at 2:15









Blue

43.6k868141




43.6k868141









asked Jul 24 at 1:48









Mattiu

759316




759316







  • 2




    Because they're not thinking like mathematicians. As you point out, you can't make (I) without making (II) at the same time. Undoubtedly, they mean a non-square rhombus, but then they should've said so.
    – Brian Tung
    Jul 24 at 2:03






  • 1




    The rhomboid they have in mind is probably the one that has 45 and 135 degree angles, with one pair of sides $sqrt2$ times the length of the other pair.
    – Brian Tung
    Jul 24 at 2:05










  • Brian, how you deduce that?
    – Mattiu
    Jul 24 at 22:05










  • Which part? If you mean the rhomboid part, I just visualized the figure shown in Parcly Taxel's answer.
    – Brian Tung
    Jul 24 at 22:34












  • 2




    Because they're not thinking like mathematicians. As you point out, you can't make (I) without making (II) at the same time. Undoubtedly, they mean a non-square rhombus, but then they should've said so.
    – Brian Tung
    Jul 24 at 2:03






  • 1




    The rhomboid they have in mind is probably the one that has 45 and 135 degree angles, with one pair of sides $sqrt2$ times the length of the other pair.
    – Brian Tung
    Jul 24 at 2:05










  • Brian, how you deduce that?
    – Mattiu
    Jul 24 at 22:05










  • Which part? If you mean the rhomboid part, I just visualized the figure shown in Parcly Taxel's answer.
    – Brian Tung
    Jul 24 at 22:34







2




2




Because they're not thinking like mathematicians. As you point out, you can't make (I) without making (II) at the same time. Undoubtedly, they mean a non-square rhombus, but then they should've said so.
– Brian Tung
Jul 24 at 2:03




Because they're not thinking like mathematicians. As you point out, you can't make (I) without making (II) at the same time. Undoubtedly, they mean a non-square rhombus, but then they should've said so.
– Brian Tung
Jul 24 at 2:03




1




1




The rhomboid they have in mind is probably the one that has 45 and 135 degree angles, with one pair of sides $sqrt2$ times the length of the other pair.
– Brian Tung
Jul 24 at 2:05




The rhomboid they have in mind is probably the one that has 45 and 135 degree angles, with one pair of sides $sqrt2$ times the length of the other pair.
– Brian Tung
Jul 24 at 2:05












Brian, how you deduce that?
– Mattiu
Jul 24 at 22:05




Brian, how you deduce that?
– Mattiu
Jul 24 at 22:05












Which part? If you mean the rhomboid part, I just visualized the figure shown in Parcly Taxel's answer.
– Brian Tung
Jul 24 at 22:34




Which part? If you mean the rhomboid part, I just visualized the figure shown in Parcly Taxel's answer.
– Brian Tung
Jul 24 at 22:34










2 Answers
2






active

oldest

votes

















up vote
1
down vote













They are using a strict definition of the three geometric shapes: a square, even though it meets the criteria for being a rhombus, is not considered one of the latter, and similarly a rhombus is not considered a rhomboid (parallelogram not a rectangle or rhombus).



That the four triangles can form a rhomboid comes from the following arrangement: If they are to form a rhombus, either the rhombus's sides are the hypotenuses of the triangles, which however would form a square, or they are twice the length of the triangles' short sides, which however does not lead to a complete rhombus.






share|cite|improve this answer





















  • I have realized that for any figure, with which a square can be formed within it, these 4 rectangular isosceles triangles can be formed, is this true? For example, it works with the rhomboid since it is a square with a triangle to the left and another to the right, it also works with the rectangle making two squares like those in the rhomboid.
    – Mattiu
    Jul 24 at 22:05

















up vote
1
down vote













It depends on the definition of a rhombus



Some say square is a rhombus, Some say a rhombus has a four equal sides with no right angles.



If the question was from textbook, it can differ



However, I think the question was unclear. it should have said



II) Rhombus that is not square



to be clear



From what I learnt, square is a rhombus, so technically your answer is correct.






share|cite|improve this answer





















  • Thanks for your answer
    – Mattiu
    Jul 24 at 22:05










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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote













They are using a strict definition of the three geometric shapes: a square, even though it meets the criteria for being a rhombus, is not considered one of the latter, and similarly a rhombus is not considered a rhomboid (parallelogram not a rectangle or rhombus).



That the four triangles can form a rhomboid comes from the following arrangement: If they are to form a rhombus, either the rhombus's sides are the hypotenuses of the triangles, which however would form a square, or they are twice the length of the triangles' short sides, which however does not lead to a complete rhombus.






share|cite|improve this answer





















  • I have realized that for any figure, with which a square can be formed within it, these 4 rectangular isosceles triangles can be formed, is this true? For example, it works with the rhomboid since it is a square with a triangle to the left and another to the right, it also works with the rectangle making two squares like those in the rhomboid.
    – Mattiu
    Jul 24 at 22:05














up vote
1
down vote













They are using a strict definition of the three geometric shapes: a square, even though it meets the criteria for being a rhombus, is not considered one of the latter, and similarly a rhombus is not considered a rhomboid (parallelogram not a rectangle or rhombus).



That the four triangles can form a rhomboid comes from the following arrangement: If they are to form a rhombus, either the rhombus's sides are the hypotenuses of the triangles, which however would form a square, or they are twice the length of the triangles' short sides, which however does not lead to a complete rhombus.






share|cite|improve this answer





















  • I have realized that for any figure, with which a square can be formed within it, these 4 rectangular isosceles triangles can be formed, is this true? For example, it works with the rhomboid since it is a square with a triangle to the left and another to the right, it also works with the rectangle making two squares like those in the rhomboid.
    – Mattiu
    Jul 24 at 22:05












up vote
1
down vote










up vote
1
down vote









They are using a strict definition of the three geometric shapes: a square, even though it meets the criteria for being a rhombus, is not considered one of the latter, and similarly a rhombus is not considered a rhomboid (parallelogram not a rectangle or rhombus).



That the four triangles can form a rhomboid comes from the following arrangement: If they are to form a rhombus, either the rhombus's sides are the hypotenuses of the triangles, which however would form a square, or they are twice the length of the triangles' short sides, which however does not lead to a complete rhombus.






share|cite|improve this answer













They are using a strict definition of the three geometric shapes: a square, even though it meets the criteria for being a rhombus, is not considered one of the latter, and similarly a rhombus is not considered a rhomboid (parallelogram not a rectangle or rhombus).



That the four triangles can form a rhomboid comes from the following arrangement: If they are to form a rhombus, either the rhombus's sides are the hypotenuses of the triangles, which however would form a square, or they are twice the length of the triangles' short sides, which however does not lead to a complete rhombus.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 24 at 2:05









Parcly Taxel

33.5k136588




33.5k136588











  • I have realized that for any figure, with which a square can be formed within it, these 4 rectangular isosceles triangles can be formed, is this true? For example, it works with the rhomboid since it is a square with a triangle to the left and another to the right, it also works with the rectangle making two squares like those in the rhomboid.
    – Mattiu
    Jul 24 at 22:05
















  • I have realized that for any figure, with which a square can be formed within it, these 4 rectangular isosceles triangles can be formed, is this true? For example, it works with the rhomboid since it is a square with a triangle to the left and another to the right, it also works with the rectangle making two squares like those in the rhomboid.
    – Mattiu
    Jul 24 at 22:05















I have realized that for any figure, with which a square can be formed within it, these 4 rectangular isosceles triangles can be formed, is this true? For example, it works with the rhomboid since it is a square with a triangle to the left and another to the right, it also works with the rectangle making two squares like those in the rhomboid.
– Mattiu
Jul 24 at 22:05




I have realized that for any figure, with which a square can be formed within it, these 4 rectangular isosceles triangles can be formed, is this true? For example, it works with the rhomboid since it is a square with a triangle to the left and another to the right, it also works with the rectangle making two squares like those in the rhomboid.
– Mattiu
Jul 24 at 22:05










up vote
1
down vote













It depends on the definition of a rhombus



Some say square is a rhombus, Some say a rhombus has a four equal sides with no right angles.



If the question was from textbook, it can differ



However, I think the question was unclear. it should have said



II) Rhombus that is not square



to be clear



From what I learnt, square is a rhombus, so technically your answer is correct.






share|cite|improve this answer





















  • Thanks for your answer
    – Mattiu
    Jul 24 at 22:05














up vote
1
down vote













It depends on the definition of a rhombus



Some say square is a rhombus, Some say a rhombus has a four equal sides with no right angles.



If the question was from textbook, it can differ



However, I think the question was unclear. it should have said



II) Rhombus that is not square



to be clear



From what I learnt, square is a rhombus, so technically your answer is correct.






share|cite|improve this answer





















  • Thanks for your answer
    – Mattiu
    Jul 24 at 22:05












up vote
1
down vote










up vote
1
down vote









It depends on the definition of a rhombus



Some say square is a rhombus, Some say a rhombus has a four equal sides with no right angles.



If the question was from textbook, it can differ



However, I think the question was unclear. it should have said



II) Rhombus that is not square



to be clear



From what I learnt, square is a rhombus, so technically your answer is correct.






share|cite|improve this answer













It depends on the definition of a rhombus



Some say square is a rhombus, Some say a rhombus has a four equal sides with no right angles.



If the question was from textbook, it can differ



However, I think the question was unclear. it should have said



II) Rhombus that is not square



to be clear



From what I learnt, square is a rhombus, so technically your answer is correct.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 24 at 2:05









Pizzaroot

1056




1056











  • Thanks for your answer
    – Mattiu
    Jul 24 at 22:05
















  • Thanks for your answer
    – Mattiu
    Jul 24 at 22:05















Thanks for your answer
– Mattiu
Jul 24 at 22:05




Thanks for your answer
– Mattiu
Jul 24 at 22:05












 

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