Geometry Challenge 144

Geometry Level 2

A B C D ABCD and B C E F BCEF are squares, where A B + D F = 18. AB+DF=18.

The area of the red region is x , x, the area of the blue region is y , y, and 3 x + 3 y = a b , 3x+3y=ab, where a a is a perfect square and b b is prime.

What is a + b ? a+b?

16 20 24 40

Vaibhav Priyadarshi by Vaibhav Priyadarshi , 17, India

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1 solution

Theodore Sinclair
Apr 15, 2018

Using a little algebra we realise that the side of a square is 6. The blue area is 1/2 the area of a square = 18 units.

Calculating the area of the green triangles we get 39. Subtract 18 to get 21. 36-21=15, the area of the red.

15 × 3 + 18 × 3 = 99 = 9 × 11 15 \times 3+18 \times 3=99=9 \times 11

9+11=20

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Can you show how you got the area of the three Green triangle pls?

Martin Karroum - 3 years, 1 month ago

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We can use coordinates to create equations for the lines. Calling the bottom left corner (0,0) and the top right (2,1) and assigning coordinates to every corner the equations for DB and AF are y=-x+1 and y=x/2 respectively. Solving these simultaneously we find the point of intersection is (2/3,1/3).

The area of Triangle with two points at A and B= ( 6 × ( 1 / 3 ) × 6 ) / 2 = 6. (6 \times (1/3) \times 6)/2=6.

The area of Triangle with two points at D and F= ( 12 × ( 2 / 3 ) × 6 ) / 2 = 24. (12 \times (2/3) \times 6)/2=24.

The area of Triangle with two points at B and E= ( 6 × ( 1 / 2 ) × 6 ) / 2 = 9. (6 \times (1/2) \times 6)/2=9.

9+6+24=39

Theodore Sinclair - 3 years, 1 month ago

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