Find the area of a parallelogram whose diagonals are 8 and 10 cm and one of the diagonals is perpendicular to the side.

| September 12, 2021 | education

Let’s draw a parallelogram ABCD, d1, d2 – diagonals. Since the perpendicular is the shortest distance between two straight lines, the smaller diagonal is perpendicular to the side (d1 is perpendicular to AB).
Distances AO = OC = AC / 2 and DO = OB = DB / 2, according to the property of the parallelogram diagonals.
Consider a triangle AOB, <B = 90` OB = BD / 2 = 4 cm; AO = AC / 2 = 5 cm.
Using the Pythagorean theorem, we find AB:
AB = ROOT (AO ^ 2-OB ^ 2) = 3 cm
The diagonal of the parallelogram divides it into 2 equal triangles:
ADB = BDC, and the area of ​​the parallelogram is equal to the sum of the areas of these triangles:
Sabc = AB * BD / 2
Sabcd = 2 * Sabc = 2 * (1/2) * AB * BD
Sabcd = AB * BD = AB * d1 = 3 * 8 = 24 cm ^ 2
Answer: 24 cm ^ 2



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