The side of the parallelogram is 21 and the diagonals are 34 and 20. Find the area of the parallelogram.

| September 25, 2021 | education

Let us draw the height DH to the AC diagonal.

Triangles ADH and CDH are rectangular. Point O divides the diagonal AC in half, then AO = CO = AC / 2 = 34/2 = 17 cm.

Let the length of the segment OH = X cm, then AH = (17 + X) cm.

From right-angled triangles АDН and СDН, according to the Pythagorean theorem, we express DH.

DH ^ 2 = AD ^ 2 – AH ^ 2 = 21 ^ 2 – (17 + X) ^ 2 = 441 – 289 – 34 * X – X ^ 2.

DH ^ 2 = DO ^ 2 – OH ^ 2 = 100 – X ^ 2.

441 – 289 – 34 * X – X ^ 2 = 100 – X ^ 2.

34 * X = 52.

X = 52/34 = 26/17.

Then DH ^ 2 = 100 – X ^ 2.

DH ^ 2 = 100 – 676/289 = 28224/289.

DH = 168/17.

Determine the area of ​​the triangle ACD.

Sacd = AC * DH / 2 = 34 * 168/17/2 = 168 cm2.

The diagonal of the parallelogram divides it into two equal triangles, then Savsd = 2 * Sasd = 336 cm2.

Answer: The area of ​​the parallelogram is 336 cm2.



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