Class 9 Exam > Class 9 Questions > If a diagonal AC and BD of a quadrilateral AB...
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If a diagonal AC and BD of a quadrilateral ABCD bisect each other, then ABCD is a
- a)Rectangle
- b)Triangle
- c)Rhombus
- d)Parallelogram
Correct answer is option 'D'. Can you explain this answer?
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Explanation:
To prove that quadrilateral ABCD is a parallelogram, we need to show that opposite sides are parallel.
Given:
- Diagonal AC and BD of quadrilateral ABCD bisect each other.
Proof:
We can prove that ABCD is a parallelogram by using the properties of diagonals in a quadrilateral.
1. Diagonals of a parallelogram bisect each other:
In a parallelogram, the diagonals bisect each other. Since the given quadrilateral has diagonals that bisect each other, it suggests that ABCD could be a parallelogram.
2. Diagonals of a parallelogram divide it into congruent triangles:
In a parallelogram, the diagonals divide it into four congruent triangles. If the given quadrilateral has diagonals that bisect each other, it implies that the resulting triangles formed by the diagonals are congruent.
3. Congruent triangles have equal corresponding angles and sides:
If the four triangles formed by the diagonals are congruent, it means they have equal corresponding angles and sides.
4. Opposite angles of a parallelogram are congruent:
In a parallelogram, opposite angles are congruent. Since the congruent triangles formed by the diagonals have equal corresponding angles, it suggests that opposite angles of ABCD are congruent.
5. Opposite sides of a parallelogram are parallel:
If opposite angles of ABCD are congruent, it implies that opposite sides are parallel. Hence, ABCD is a parallelogram.
Therefore, based on the properties of diagonals in a quadrilateral, we can conclude that if diagonal AC and BD of quadrilateral ABCD bisect each other, then ABCD is a parallelogram (option D).
To prove that quadrilateral ABCD is a parallelogram, we need to show that opposite sides are parallel.
Given:
- Diagonal AC and BD of quadrilateral ABCD bisect each other.
Proof:
We can prove that ABCD is a parallelogram by using the properties of diagonals in a quadrilateral.
1. Diagonals of a parallelogram bisect each other:
In a parallelogram, the diagonals bisect each other. Since the given quadrilateral has diagonals that bisect each other, it suggests that ABCD could be a parallelogram.
2. Diagonals of a parallelogram divide it into congruent triangles:
In a parallelogram, the diagonals divide it into four congruent triangles. If the given quadrilateral has diagonals that bisect each other, it implies that the resulting triangles formed by the diagonals are congruent.
3. Congruent triangles have equal corresponding angles and sides:
If the four triangles formed by the diagonals are congruent, it means they have equal corresponding angles and sides.
4. Opposite angles of a parallelogram are congruent:
In a parallelogram, opposite angles are congruent. Since the congruent triangles formed by the diagonals have equal corresponding angles, it suggests that opposite angles of ABCD are congruent.
5. Opposite sides of a parallelogram are parallel:
If opposite angles of ABCD are congruent, it implies that opposite sides are parallel. Hence, ABCD is a parallelogram.
Therefore, based on the properties of diagonals in a quadrilateral, we can conclude that if diagonal AC and BD of quadrilateral ABCD bisect each other, then ABCD is a parallelogram (option D).
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If a diagonal AC and BD of a quadrilateral ABCD bisect each other, then ABCD is aa)Rectangleb)Trianglec)Rhombusd)ParallelogramCorrect answer is option 'D'. Can you explain this answer?
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