Answer :
Certainly! Let's break down the problem step-by-step.
### Given Statement:
- p: A figure is a square.
- q: The figure’s diagonals divide it into isosceles triangles.
- The given statement can be written as p → q, which means: "If a figure is a square, then its diagonals divide it into isosceles triangles."
### Converse of the Statement:
- The converse of the statement p → q is q → p.
- q → p means: "If a figure's diagonals divide it into isosceles triangles, then the figure is a square."
### Truth Value of the Converse:
To determine whether the converse q → p is true, we need to assess if every figure that has diagonals dividing into isosceles triangles is necessarily a square.
A square certainly has diagonals that cut it into isosceles right triangles. However, a general quadrilateral, such as a rectangle, also has diagonals that bisect each other and form isosceles triangles (though not necessarily right triangles unless it is a square).
Therefore, just because a figure's diagonals divide it into isosceles triangles does not strictly mean the figure must be a square. It could potentially be another type of quadrilateral as well.
### Consolidating the Result:
Given this information, we conclude that the converse q → p is not always true. It can be true in some cases (as when the figure is indeed a square), but there are cases where it is not true (such as other types of quadrilaterals with the same property).
Therefore, we can state:
"The converse of the statement is sometimes true and sometimes false."
### Final Answer:
The correct representation is:
"The converse of the statement is sometimes true and sometimes false."
### Given Statement:
- p: A figure is a square.
- q: The figure’s diagonals divide it into isosceles triangles.
- The given statement can be written as p → q, which means: "If a figure is a square, then its diagonals divide it into isosceles triangles."
### Converse of the Statement:
- The converse of the statement p → q is q → p.
- q → p means: "If a figure's diagonals divide it into isosceles triangles, then the figure is a square."
### Truth Value of the Converse:
To determine whether the converse q → p is true, we need to assess if every figure that has diagonals dividing into isosceles triangles is necessarily a square.
A square certainly has diagonals that cut it into isosceles right triangles. However, a general quadrilateral, such as a rectangle, also has diagonals that bisect each other and form isosceles triangles (though not necessarily right triangles unless it is a square).
Therefore, just because a figure's diagonals divide it into isosceles triangles does not strictly mean the figure must be a square. It could potentially be another type of quadrilateral as well.
### Consolidating the Result:
Given this information, we conclude that the converse q → p is not always true. It can be true in some cases (as when the figure is indeed a square), but there are cases where it is not true (such as other types of quadrilaterals with the same property).
Therefore, we can state:
"The converse of the statement is sometimes true and sometimes false."
### Final Answer:
The correct representation is:
"The converse of the statement is sometimes true and sometimes false."
