The perimeter of the rhombus is 68, the area of the rhombus is 240. Find the diagonals of the rhombus.

| September 29, 2021 | education

Since the lengths of the sides of the rhombus are equal, then AB = BC = CD = AD = Ravsd / 4 = 68/4 = 17 cm.

Let the length of the smaller diagonal be 2 * X cm, and the length of the larger diagonal is 2 * Y cm.

Then OB = 2 * X / 2 = X cm, AO = 2 * Y / 2 = Y cm.

The area of ​​the rhombus is equal to: Savsd = AC * BD / 2 = 2 * Y * 2 * X / 2.

2 * X * Y = 240.

X * Y = 120. (1)

In a right-angled triangle AOB, according to the Pythagorean theorem:

AB ^ 2 = AO ^ 2 + OB ^ 2.

289 = Y ^ 2 + X ^ 2. (2).

Let’s solve the system of equations 1 and 2.

Y = 120 / H.

289 = (120 / X) ^ 2 + X ^ 2.

289 * X ^ 2 = 14,400 + X ^ 4.

X ^ 4 – 289 * X ^ 2 + 14,400 = 0.

Let X ^ 2 = Z, then Z ^ 2 – 289 * Z + 14400 = 0.

Let’s solve the quadratic equation.

Z1 = 64, Z2 = 225.

X1 = 8 cm, X2 = 15 cm.

Y1 = 120/8 = 15 cm.

Y2 = 120/15 = 8 cm

Then, if BD = 2 * 8 = 16 cm, AC = 2 * 15 = 30 cm.

If BD = 2 * 15 = 30 cm, AC = 2 * 8 = 16 cm.

Answer: The diagonals of the rhombus are 16 cm and 30 cm.



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