Geometry Problem Solving

Below are various problems that explore topics discussed in geometry. These problems may be solved on paper or by using Geometer's Sketchpad. The problems are meant to enhance problem solving and encourage critical thinking. When working these problems be sure that you are able to justify your answers. Enjoy!

#1

How large is the largest triangle that can be placed inside a unit square?

Hint: Consider using Geometer's Sketchpad to justify your answer.

What is meant by "largest"? Does it refer to length of side or area?

Would the largest triangle when referring to the length of a side be a different triangle from one referring to area?

Check your work using Geometer's Sketchpad.

#2

Find the coordinates of a point that is five units from (0, 0).

How many more points can you find that satisfy this condition and have integral coordinates?

How many point would there be if the coordinates do not have to be integral? If P is a point with coordinates (x, y) that is five untis from (0, 0), what relationship must exist between x, y and 5?

A GSP Sketch

Hints

Let's continue with this line of thinking.....

Find a point on the x-axis that is 5 units away from (2, 3). How many are there?

Find a point on the line x=7 that is 5 units from the point (2, 3). Again, how many are there?

Now that we know how to write an equation of a circle or sketch the circle given its equation, what happens if you are not given an equation but some other information sufficient to determine the circle?

Notice in the picture above you are given 3 points that lie on a unique circle. Find the equation of the circle.

Have you figured it out yet? Check your answer here.

#3

A surveyor needed to determine the area of a plot of land labeled ABCDE diagrammed below. He located the north-south line through E and the east-west lines through A, B, C, and D. He found that AO=37 m, BR = 47 m, CQ = 42 m, DP = 28 m, PQ = 13 m, QE = 7 m, ER = 19 m, and RO = 18m. Find the required area using the surveyor's measurements.

After you have solved this problem click here to check your answers.

Could the surveyor used another method to solve the problem? Explain.

#4

Why are there no giants?

Giant are found in many movies and books. From Jack and the Beanstalk to Godzilla and the giant Marshmellow Man in Ghostbusters, our imagination allows for extremely large beings.

In the movie, Honey I Blew Up the Kid, an eccentric scientist performs a wacky experiment which causes his toddler to eventually grow to a height of 112 feet while maintaining a similar body shape.

If the average toddler is 2 1/2 feet tall, consider:

a) If the hand span of an average toddler is 4 inches, what would the hand span of the 112 feet giant kid be?

b) When the average toddler goes on spring break to the beach suppose he uses 2 ounces of sunscreen. How much suntan lotion would he require as giant 112 foot toddler?

c) Suppose the average toddler drinks 24 ounces of grape juice a day. How much juice could would the giant kid drink?

d) Give a good argument for what you think the giant toddler would weigh.

e) Based upon the questions addressed above, where do movie directors violate geometrical properties?

Use GSP to expore to situation.

Some introductory references and websites to explore:

The Surface Area and Volume Ratio

The sizes of living things

Ask Dr. Math

Application with water spiders

Application to health

Extension Reference:

Surface Area to Volume Ratio of Platonic Solids:

Dodecahedron

#5

Find the area of the largest circle that can be inscribed in an equilateral triangle of side 12 cm.

Hint: Consider using Geometer's Sketchpad to justify your answer.

a) How is the point E found in order to construct the inscribed circle?

b) What is the relationship between the length of the segment AE and the length of the segments used to form point E?

c) What is the radius of the circle expressed as an exact answer?

#6

Given an equilateral triangle ABC, such that AB is a diameter of a given circle and the circle intersects the other two sides of the triangle at points D and E. If the diameter of the circle is 16, find the area of the inscribed quadrilateral ABED.

HINT: In order to solve the problem, draw the figure and use what you know about triangles and circles to find the necessary measurements.

Click here for an explanation of the problem.

#7

Romeo has a ladder which is 32 feet long when fully extended. He needs to reach a window sill that is 30 feet up the wall from the ground to elope with Juliett. If he can not place the bottom of the ladder any closer than 10 feet from the castle due to the moat, will the top of the ladder reach the window sill?

Explain how you came to your decision.

Click here for GSP Sketch

If the bottom of the ladder is moved 2 feet from the its position above, does the ladder drop 2 feet down from the wall?

#8 What is the sum of angles A + B + C + D + E?

Write an informal proof to explain your answer.

Link to GSP for exploration.

The Proof.

#9

At the airline terminal, a circular waiting area is built with a circular ticket booth in the middle. The two circles are concentric and the ticket booth (shaded area) extends all the way to the ceiling. The diameter of the ticket booth is 20 meters and the diameter of the room itself (if the ticket booth was removed) is 48 meters. If I sit at point P against the outer wall, what is the longest distance I can see in the room?

Use Geometer's Sketchpad to explore this problem.

#10

If A cow is tethered to a barn that is 40 meters by 40 meters with a rope 80 meters long. Over how large an area (measured in square meters) can the cow graze? Note that the fence on the right side is only 30 meters long while the fence on the left is at least 100 feet.

Try to examine this problem with pencil and paper first.

Use Geometer's Sketchpad to explore this problem.

#11

A circle with center O intersects a circle with center P at two points A and B. The circles have different radii. The line segment AB (called the common chord) intersects the line connecting the centers O and P at R.

What can be proven in the diagram? (I suggest you draw a diagram first!)

I need a hint!

#12

Pipes are often transported by stacking three together and wrapping them with a metal band. If each of the pipes has an outside diameter of 12 inches, how long must the metal band be to completely encircle all three pipes once?

If you don't know where to begin use a GSP sketch with three tangent congruent circles.

Still don't know what to do or how to begin? Here's a good beginning. Hint #1

Still stuck? Hint #2

Check your answer after you have attempted the problem using the above hints.

How did you do?

#13

ABCD is a square with side lengths of 1. The squares inside ABCD are formed by connecting the midpoints of the larger square.

What happens to the perimeter of each successive square?

Calculate the sum of the perimeters of the first eight squares (four are shown). If this reiteration is continued indefinitely, will the sum of the perimeters approach infinity or approach a specific value? Why?

Go GSP for a hint.

I'm confused. I need more help!

Is the figure below a regular polygon? Justify your answer.

Wait! There is more to come!

Porfolio Questions to Ponder....

.....and discuss, explain, illustrate, prove, expand, seek counter examples,

or simply think about on paper.

When a piece of paper if folded, why is the crease straight?

Why is it that chairs so often rock from one leg to another while three legged stools never have this problem?

Is it possible to cut an obtuse triangle into triangular pieces, all of which are acute?

Our principal decided to plant an orchard of 10 trees (in place of the portable classrooms). However, being a student of mathematics at heart, he set the trees out in five rows with four trees in each row. Can you determine how he did this?

Explain why the line through (0,3) and (4,6) is said to be parallel to the line that contains the points (-6, -2) and (-2,1).

How can you measure the width of a river without really crossing it?

Using your Geometry book for the definitions of parallelogram, rectangles, rhombus and square, make a diagram to illustrate how these figures relate to each other. Place a trapezoid and kite in this diagram and justify the its position.

Plot the points A(2,3), B(7,4), and C(9,8). Find a point D such that ABCD is a parallelogram. Also find another point that would make CABD a parallelogram. Is is possible to make a parallelogram which could be labeled ACBD?

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