Re: In the right triangle ABC shown above, BDEF is a square.
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28 Jun 2023, 08:40
First notice that this is a 45,45, 90 triangle. So the second side will be 6.
We do not know if the square is the midsegement of the legs of the triangle, but nonetheless we can find out its length
Let the sides of the square be x.
That means we have cut the triangle into 3 shapes: the square x, and two triangles - one with base x, and the height (6-x) and another with height x and base 6-x.
The area of the triangle is simply given by (1/2)(6^2)=18.
Knowing the area of the big triangle, we know that the 3 shapes inside must equal the same area.
Therefore:
\(18=0.5(6-x)(x)+0.5(x)(6-x)+x^2\) (area of two triangles+square)
\(18=(6-x)(x)+x^2\)
Now, becuase this is a quantity comparison question we can simply use the answers - QB is 9. So if the area of the square were 9, that implies that x is 3. If the resulting sum is less than 18, that means x is greater than 3 and QA must be bigger. If the resulting sum is equal, then they are the same, and if the resulting sum is greater than 18 x must be smaller than 3 so QB is bigger.
\(18=(6-3)(3)+3^2=3*3+3^2=9+9=18\)
We get 18 on the dot, so that means that x is 3, and the two quantities must be the same.
C