The perimeter of a quadrilateral circumscribed about a circle is 56 cm.

| May 26, 2021 | education

The perimeter of a quadrilateral circumscribed about a circle is 56 cm. Find the sides of a quadrilateral, keeping in mind that two of its sides are 2: 3 and the other two are 5: 8

Two sides of the quadrangle are related as 2: 3, and the other two – as 5: 8. Let’s designate the first pair of sides as 2a and 3a, the second – as 5b and 8b.

Since the quadrangle is described, the sums of its opposite sides are equal, that is, each pair of opposite sides is 56: 2 = 28 cm.

This means that the system is obtained: 2a + 8b = 28; 3a + 5b = 28.

Multiply the first equation by 3, and the second by 2:

2a + 8b = 28 (* 3), 6b + 24b = 84.

3a + 5b = 28 (* 2), 6a + 10b = 56.

Subtract the second from the first equation: (6a + 24b) – (6a + 10b) = 84 – 56.

24b – 10b = 28;

14b = 28;

b = 2.

This means that side 5b = 5 * 2 = 10 cm, and side 8b = 8 * 2 = 16 cm.

Substitute the value b = 2 into any equation:

2a + 8b = 28; 2a + 8 * 2 = 28; 2a + 16 = 28; 2a = 28 – 16; 2a = 12; a = 6.

This means that side 2a = 2 * 6 = 12 cm, and side 3a = 3 * 6 = 18 cm.

Answer: the sides of the quadrangle are 10 cm, 16 cm, 12 cm and 18 cm.



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