Geometry Worksheet Template printable pdf download

GEOMETRY PROBLEMS

via Complex Numbers and Geometric Transformations

Berkeley Math Circle – Advanced

by Zvezdelina Stankova

Berkeley Math Circle Director

November 23, 2010

1. Preparatory Plane Geometry Problems given to BMC-Intermediate

Problem 1. (Three Squares) Three identical squares with bases AM , M H, and HB are put next

to each other to form a rectangle ABCD. Find the sum of the angles

AM D + AHD + ABD

and prove that your answer is correct.

Problem 2. (Research Problem) Generalize Problem #1 to 4 squares. Can you ﬁnd the sum

of the resulting four angles? How about the same problem for 5 squares? How about n squares?

What happens when we let n go to inﬁnity (i.e., we use an inﬁnite number of squares): will the

sum of the angles be a

angle, or will all angles add up to

Problem 3. (Farmer and Cow) During a hot summer day, a farmer and a cow ﬁnd themselves

on the same side of a river. The farmer is 2 km from the river and the cow is 6 km from the river.

If each of them would walk straight to the river, they would ﬁnd themselves 4 km from each other.

Unfortunately, the cow has broken its leg and cannot walk. The farmer needs to get to the river,

dip his bucket there, and take the water to the cow. To which point on the river should the farmer

walk so that his total walk to the river and then to the cow is as short as possible?

Problem 4. (Shortest Broken Line) Two lines p

and p

intersect. Two points A and B lie in

the acute angle formed by the lines. Find a point C on p

and a point D on p

so that the broken

line ADBCA has the smallest possible length. Prove that the points you have found indeed yield

this smallest possible length.

Problem 5. (Minimal Perimeter) Given

ABC, on the ray opposite to ray CA take a point

so that CB

= CB . Prove that

(a) Point B

is the reﬂection of B across the angle bisector l of the exterior angle of the triangle

at vertex C.

(b) If D is an arbitrary point on l diﬀerent from C, the perimeter of

ABD is bigger than the

perimeter of

ABC.

Problem 6. (Find the Perimeter) In

ABC, AC = BC and AB = 10 cm. Through the

midpoint D of AC we draw a line perpendicular to AC. This line intersects BC in point E. The

perimeter of

ABC is 40 cm. Find the perimeter of

ABE.

Problem 7. (Locating Angle Bisector) Two points A and B and a line l are given so that the

line intersects segment AB (neither A nor B lies on l). Find point C on l so that the angle bisector

ACB lies on l. Prove that your construction is correct.

Geometry Worksheet Template

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