Solution: Australian Mathematics Competition 2022 – Upper Primary, Problem 16

Let's discuss a problem from AMC 2022 - Upper Primary - 16 on Geometry.

Problem

Three different squares are arranged as shown. The perimeter of the largest square is $32$ centimetres. The area of the smallest square is $9$ square centimetres. What is the perimeter of the medium-sized square? (AMC 2022 - Upper Primary - 16)

(A) 12 cm (B) 14 cm (C) 20 cm (D) 24 cm (E) 30 cm

Solution

In this problem, it is given that the big square has a perimeter of \(32\)cms. We know how to find the perimeter of a square.
The perimeter is nothing but the length of the boundary. So \(4a\) is the perimeter of a square. Thus \(4a = 32\),

so, \(a = \frac {32}{4} = 8\).

Thus the side length of the bigger square is = \(8\)cms.

The small square has the area as \(9\) sq cms.

The area of a square is = \( a^2\). Thus \(a^2 = 9, a = 3\).

Thus the side length of the smallest square is 3 cms.

Thus the medium square has a side length of = \(8-3\) cm = \(5\)cm.

Thus the perimeter of the medium square is = \(4a = 4 \times 5 = 20\) cm.

What is AMC (Australian Mathematics Competition)?

The Australian Mathematics Competition (AMC) is one of the largest and oldest annual mathematics competitions in Australia, aimed at fostering interest and excellence in mathematics among students