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A figure is composed of four congruent right triangles.
If the area of the large square is 13 square units, the area of the small square in
the middle part is 1 square unit, then what is the sum of the cubes of the two legs of each right triangle?

A Figure Is Composed Of Four Congruent Right Triangles If The Area Of The Large Square Is 13 Square Units The Area Of The Small Square In The Middle Part Is 1 S class=

Sagot :

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SOLUTION:

Let x and y be the two legs of each right triangle.

As shown in the figure, the area of the small square in the middle part is 1 square unit, and the area of the large square is 13 square units.

Thus, the sum of the four congruent right triangles is 13 - 1 = 12 square units.

Now, add a right triangle with same area to the outside of each right triangle such that this forms another larger square.

Hence, the area of the larger square is 13 + 12 = 25 square units.

This implies that the side of the square is 5 units as s² = Area.

Moreover, we can observe that the sum of the two legs of the right triangle is equal to the side length of the largest square, which is 5 units.

Also, the difference between the two legs is exactly equal to the side length of the original square; that is 1 unit.

So,

[tex]\begin{cases} \sf x + y = 5\\ \sf x - y = 1\\\end{cases}[/tex]

Adding the two equations and solving, we get (x,y) = (3,2).

Therefore, the sum of the cubes of the two legs of each right triangle is

3³ + 2³ = 35 units

ANSWER:

35 units.

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