# The perimeter of the rectangle is 34 and its diagonal is 13, find the sides of the rectangle

September 30, 2021 | education

| **The perimeter of the rectangle is 34 and its diagonal is 13, find the sides of the rectangle, the problem is solved by the system.**

Let the first side be x cm, and the second side y cm.

Let’s compose a system of equations:

2x + 2y = 34;

x ^ 2 + y ^ 2 = 132;

Let’s simplify the first equation:

x + y = 17.

x = 17 – y.

Substitute the value of x in the equation x ^ 2 + y ^ 2 = 169.

(17 – y) ^ 2 + y ^ 2 = 169;

172 – 2 * 17 * y + y ^ 2 + y ^ 2 = 169;

289 – 34y + 2y ^ 2 = 169;

y ^ 2 – 17y + 60 = 0;

We have a = 1, b = – 17, c = 60.

D = b ^ 2 – 4ac = (-17) ^ 2 – 4 * 60 = 289 – 240 = 49.

x1 = (17 + 7): 2 = 12 cm.

x2 = (17 – 7): 2 = 5 cm.

Hence y1 = 17 – 12 = 5 cm.

y2 = 17 – 5 = 12 cm.

Answer: the sides can be 12 and 5 cm or 5 and 12 cm.

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