0

Here is a cute little problem.

In the diagram below $ABC$ is a right triangle, $\angle ABC$ is a right angle. The blue region inside the triangle is a rectangle. Given that point $G$ is both a corner of the rectangle and the centroid of the triangle, and that the area of the blue region is $24$ unit square, find the area of the triangle.

enter image description here

(Below is one possible approach. Can you come up with something different?)

enter image description here

blackened
  • 1,115

2 Answers2

1

@MichaelRozenberg has answered the post already. I am just adding the details.

The ratios dividing the lines are as shown.

enter image description here

[X] = … = 12

[Y] = [X]/2 = 6

Also, $[Y] = (\dfrac {1}{1+2})^2 \times [\triangle ABD] = \dfrac 19 \times (\dfrac 12 [\triangle ABC])$

Then, $[\triangle ABC]) = ....$

Mick
  • 17,141
0

$$\Delta ABC=\frac{1}{2}AB\cdot BC=\frac{9}{2}\cdot\frac{AB}{3}\cdot\frac{BC}{3}=\frac{9}{2}\cdot24=108.$$

Michael Rozenberg
  • 194,933
  • So how do you know the sides of the triangles divided in thirds? Do not answer in Comments, please add this explanation to the answer. – Oscar Lanzi Jun 09 '19 at 12:07