F

A______B______E

C______D

F

A______B______E

C______D

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I'm going to do this based on the way I think your drawing should've looked (as it is now, ACEF does not make a square...). It seems like it should look like this, based on your tag about congruent triangles.

F

A B E

D C

If you draw this on paper, you'll see it breaks up into 5 congruent triangles. The
**area of square ABCD is twice the area of triangle ABC,** and the area of triangle ACEF is four times the area of triangle ABC. Area of a triangle is half the base times the height.

The area of triangle ABC: 0.5*AB*BC

Area of square ABCD: 2*0.5*AB*BC but we know AB=BC as it's a square. So **
Area of square ABCD = 2*0.5*AB*AB = AB^2 **

AB = 1/2*AE, and the **area of ACEF is twice the area of triangle ACE**.

Area of triangle ACE = 0.5*AE*BC, but BC=AB and AE = 2*AB, so area of ACE = 0.5*2*AB*AB = AB^2

**Area of square ACEF = 2*AB^2** which is twice the area of ABCD, so the area of ABCD is half the area of ACEF.

Statement Reason

ABCD is a square Given

AECF is a square Given

AB = (1/2)AE Given

Area ABCD = AB·AC Def. of area of a square

Area AECF = AE·AC Def. of area of a square

Area ABCD = [(1/2)AE]·AC Substitution

Area of ABCD = (1/2) AE·AC Substitution

## Comments

You could also prove it using pythagoras theorem and the fact that the area of a square is equal to the side length squared.

Area of ABCD = AB^2You could also write the area of ABCD as (using AB = 0.5*AE thus AE = 2*AB): (0.5*AE)^2 = (0.5*2*AB)^2 = AB^2

Area of ACEF = AC^2Use pythag to find length of AC: AC^2 = AB^2+BC^2 = AB^2+AB^2 = 2*AB^2

Note that the

area of ACEF = AC^2 = 2*AB^2 = 2*(area of ABCD)so the area of ABCD is half that of ACEF.