Deduction 8 - What is the point of intersection of the two diagonals in a parallelogram?

Transcript

Deduction 8 What is the point of intersection of the two diagonals in a parallelogram?We need to find out if Diagonals bisect each other To do this, we consider a Parallelogram EASY To check sides, we use congruency We prove triangles ∆ AOE & ∆ YOS congruent Since ABCD is a parallelogram Opposite sides are equal ∴ AE = YS Opposite sides of Parallelogram are parallel So, AE & YS are parallel And, AY is a transversal Since Alternate Angles are equal ∴ ∠ EAO = ∠ SYO Similarly, AE & YS are parallel And, ES is a transversal Since Alternate Angles are equal ∴ ∠ AEO = ∠ YOS In ∆ AOE and ∆ YOS ∠ EAO = ∠ SYO ∠ AEO = ∠ YOS AE = YS ∴ ∆ AOE ≅ ∆ YOS By CPCT (Corresponding Parts of Congruent Triangles) OA = OY OE = OS Thus, we can say O is mid-point of AY O is mid-point of ES This shows that the Diagonals of a parallelogram bisect each other Note: Diagonals of a parallelogram are not always equal, but they bisect each other