A square is inscribed in triangle ABC so that its two vertices lie on the side AB of one vertex – on the sides AB and BC

A square is inscribed in triangle ABC so that its two vertices lie on the side AB of one vertex – on the sides AB and BC. Find the area of the square if AD = 40 cm and the height drawn from the vertex C is 24 cm long.

An error was made in the problem statement. AB = 40 cm, not AD = 40 cm.

1. Vertices of the square M, K, P, T. Points M and K on the AB side, P on the BC side, T on the AC side.

CE is the height of the triangle ABC. CO is the height of the PCT triangle.

2. Triangles ABC and PCT are similar in two equal angles. ∠А – general. ∠РTС = ∠A as appropriate.

3. Take the length of the side PT of the PCT triangle as x (cm). РT is the side of the MCRT square.

4. Let’s make the proportion:

AB / РT = CE / CO.

CO = CE – x = (24 – x).

40 / x = 24 / (24 – x).

24x = 960 – 40x.

64x = 960.

x = 15 cm.

РT = 15cm.

5. We calculate the area of ​​the square of the MCРT:

15 x 15 = 225 cm².

Answer: the area of ​​the square of the MCРT is 225 cm².