A rhombus is a quadrilateral that has four congruent sides. How would you prove that the diagonals of a rhombus intersect at a point that is the midpoint of each diagonal?

Answer 1

See the proof below

Assume a rhombus #ABCD# with diagonals #AC# and #BD# intersecting at point #O#.
Triangles #Delta ABD# and #Delta CBD# are congruent by three sides. Therefore, angles #/_ABD# and #/_CBD# are congruent.
Consider now triangles #Delta ABO# and #Delta CBO#. They are also congruent since #AB~=CB#, #BO# is common and angles #/_ABD# and #/_CBD# are congruent (side-angle-side). Therefore, #AO~=CO#.
Similarly #BO~=DO# if you compare triangles #Delta BAC# and #Delta DAC#, then angles #/_BAC# and #/_DAC# and, finally, triangles #Delta BAO# and #Delta DAO#.
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Answer 2

To prove that the diagonals of a rhombus intersect at a point that is the midpoint of each diagonal, we can use properties of parallelograms.

  1. Label the vertices of the rhombus as A, B, C, and D.
  2. Draw the diagonals AC and BD.
  3. Since a rhombus is a special type of parallelogram, its opposite sides are parallel and congruent.
  4. Consider triangle ABC. Since AB = BC (due to the definition of a rhombus), and angle ABC = angle BCA (opposite angles of a parallelogram are congruent), triangle ABC is isosceles.
  5. By the properties of isosceles triangles, the altitude from vertex B will bisect side AC, and similarly, the altitude from vertex C will bisect side AB.
  6. Therefore, the point of intersection of the diagonals (point E) is equidistant from vertices A and C, and from vertices B and D.
  7. Hence, point E is the midpoint of both diagonals AC and BD, as required.

Therefore, we have proved that the diagonals of a rhombus intersect at a point that is the midpoint of each diagonal.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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