In a right-angled triangle with legs 18 units. and 9 units. a square is inscribed that has a common right

In a right-angled triangle with legs 18 units. and 9 units. a square is inscribed that has a common right angle with the triangle. Calculate the perimeter of the square.

1. Let’s designate the vertices of the triangle ABC, angle C = 90 °. Vertices of the square CKML. AC = 18 units, BC = 9 units.

2. Triangles ABC and AKM are similar, since they have two equal angles: angle A is common, angles AMK and ABC are equal as corresponding with parallel straight lines KM and BC and secant AB. This means that the relations of similar sides are equal: AK / AC = KM / BC.

3. Take the sides of the square as x.

5. Let’s compose the equation:

(18-x) / 18 = x / 9:

27x – 162:

x = 6.

6. The perimeter of the square is 6 x 4 = 24 units.