Geometry Challenge 37

Geometry Level 2

$ABCD, DCFE,$ and $BCHG$ are squares.

If $EF + BF = 15,$ what is the area of the red region?

The answer is 20.

4 solutions

Ahmed Almubarak
Apr 28, 2018

Rab Gani
May 9, 2018

Use coordinate geometry with G(0,0), and side of 5. AF is perpendicular to GD. So the Eqs. of lines GD:y=2x, AF: y=-1/2 x + 10, then the intersection point (4,8). The red area = 20.

Edwin Gray
Jun 26, 2018

The equation of the line AF is y = -(1/2)x + 10. The equation of the line GD is y = 2x. If P is the point of intersection, then P = (4,8) Since the product of slopes = -1, the lines are perpendicular and the red area is (1/2)(GP)(AP). Knowing the coordinates, the distances are readily calculated. AF = sqrt(20(. GP = sqrt(80). A = (1/2)(AF)(GP) = 20. Ed Gray

Vedant Saini
May 19, 2018

Let the point of intersection of GD and FA be I

Note that angle EAF and angle DGA are equal Also angle GDA and angle FAB are equal Therefore, by AA similarity of triangles, triangle GIA and triangle DIA are similar

We know that the side of the squares in the diagram is 5 Notice AG = 2 x AD Therefore the corresponding sides of triangles GIA and DIA are in the ratio 1:2 Which implies that their areas must be in the ratio 1:4

We know that the total area of triangles GIA and DIA is 25 (it is half of the area of 2 squares which is 50) Putting area of triangle DIA = x and the area of triangle GIA = 4x, We get x + 4x = 25 5x = 25 x = 5 Thus, the area of triangle GIA is 4x = 4 times 5 = 20

Regards Vedant

Geometry Challenge 37

The answer is 20.

4 solutions

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