Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Ok
To prove that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus, we can use the properties of a rhombus and the definitions of bisecting diagonals.
Let's consider a quadrilateral ABCD with diagonals AC and BD intersecting at point O, such that they bisect each other at right angles.
To show that ABCD is a rhombus, we need to prove that all four sides are congruent.
Proof:
1. Since AC and BD bisect each other at right angles, we can state that angles AOB, BOC, COD, and DOA are all right angles. This is given.
2. Now, we will prove that opposite sides of ABCD are parallel.
a. Considering diagonal AC, it divides the quadrilateral ABCD into two triangles, namely, triangle AOC and triangle COD.
b. According to the property of diagonals in a quadrilateral, if the diagonals bisect each other, then the opposite triangles formed are congruent.
c. Therefore, triangle AOC is congruent to triangle COD by the property of diagonal bisecting.
d. By the corresponding parts of congruent triangles (CPCTC), we can conclude that AO is congruent to CO.
e. Similarly, considering diagonal BD, we can prove that BO is congruent to DO.
f. Since opposite sides AO and CO, as well as sides BO and DO, are congruent, we can conclude that opposite sides of ABCD are parallel.
3. Now, let's prove that all four sides are congruent.
a. Considering triangle AOB, we have right angles at O and A.
b. Since opposite sides AO and BO are congruent (from step 2e), we can apply the Pythagorean theorem to triangle AOB to show that AO^2 + BO^2 = AB^2.
c. Similarly, considering triangle COD with right angles at O and C, we can use the Pythagorean theorem to show that CO^2 + DO^2 = CD^2.
d. Since AO^2 + BO^2 = CO^2 + DO^2 (as the diagonals bisect at right angles), we can conclude that AB^2 = CD^2, which implies that AB = CD.
e. Similarly, we can prove that BC = AD.
4. Therefore, we have shown that opposite sides of ABCD are congruent, which satisfies the definition of a rhombus.
5. Hence, we can conclude that if the diagonals of a quadrilateral bisect each other at right angles, the quadrilateral is a rhombus.
In this proof, we used properties of diagonals and congruent triangles as well as concepts from Euclidean geometry such as the Pythagorean theorem.