2020 AMC 10B Problem 21

Question

In square

ABCD,

points

E

and

H

lie on

\overline{AB}

and

\overline{DA},

respectively, so that

AE=AH.

Points

F

and

G

lie on

\overline{BC}

and

\overline{CD},

respectively, and points

I

and

J

lie on

\overline{EH}

so that

\overline{FI} \perp \overline{EH}

and

\overline{GJ} \perp \overline{EH}.

See the figure below. Triangle

AEH,

quadrilateral

BFIE,

quadrilateral

DHJG,

and pentagon

FCGJI

each has area

1.

What is

FI^2?

\dfrac73

8-4\sqrt2

1+\sqrt2

\dfrac74\sqrt2

2\sqrt2

Answer: B

Concepts:square (geometry)area decomposition special right triangle

Difficulty rating: 1950

Small hint:

The four given regions fill the square, so the square’s side length is known

Big hint:

Extend the perpendicular through

F

to meet

AB,

and compare the two resulting right isosceles triangles

Video solution:

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Written solution:

The four named regions fill the square and each has area

1,

so the square has area

4

and side length

2.

Since triangle

AEH

is right isosceles with area

1,

we have

AE=AH=\sqrt2.

Extend

FI

to meet

AB

at

K,

and set

x=BF

and

t=BE=2-\sqrt2.

Because

EH

has slope

-1,

line

FK

has slope

1,

so

BF=BK=x.

If

K

were on segment

EB,

then region

BFIE

would lie inside triangle

BFK,

whose area would be at most

\frac{t^2}{2}<1,

a contradiction. Thus

K

lies to the left of

E,

and

EK=x-t.

Triangle

BFK

is right isosceles with area

\frac{x^2}{2}.

Triangle

EIK

is right isosceles with hypotenuse

EK=x-t,

so its area is

\frac{(x-t)^2}{4}.

Since their difference is region

BFIE,

1=\frac{x^2}{2}-\frac{(x-t)^2}{4}.

Therefore

\begin{aligned} 4&=2x^2-(x-t)^2\\ &=(x+t)^2-2t^2. \end{aligned}

Also,

FK=x\sqrt2

and

KI=\frac{x-t}{\sqrt2},

so

FI=FK-KI=\frac{x+t}{\sqrt2}.

It follows that

\begin{aligned} FI^2&=\frac{(x+t)^2}{2}\\ &=2+t^2\\ &=2+(2-\sqrt2)^2\\ &=8-4\sqrt2. \end{aligned}

Thus, the correct answer is B .

2020 AMC 10B Problem 21

Answer: B

Difficulty rating: 1950

Small hint:

Big hint:

Video solution:

Written solution:

Problem 21 in other years