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Pythagorean Theorem

Problems in this category are designed for the use of the Pythagorean
theorem.
For problems which use the properties of special right triangles, see the special right triangles category.
 Related Resources

Interactive resources from our Math Tools project:
Geometry: Pythagorean Theorem
The closest match in our Ask Dr. Math archives:
Pythagorean Theorem Proofs
NCTM Standards:
Geometry Standard for Grades 912
Access to these problems requires a Membership.

Ants Marching
 Annie Fetter

Geometry, difficulty level 2. Traveling along the edges of a cross section of the cube, find the
length of the shortest possible path from one corner of a cube to the
opposite corner, as well as the longest possible path between the same
corners.
... more>>

Area of a Rectangle
 Annie Fetter

Geometry, difficulty level 2. Given rectangle ABCD with point E on CD. AE is 3, BE is 4, and AE is perpendicular to BE. What's the area of ABCD?
... more>>

As Far as the Eye Can See
 Annie Fetter

Geometry, difficulty level 4. If I'm 5'10" tall, can I see the entire length of Lake Memphramagog, which is about 23 miles long?
... more>>

Boxing Up Harry's Broom
 Annie Fetter

Geometry, difficulty level 1. Help Harry find the right size box for his broom.
... more>>

Building a Dog Enclosure
 Annie Fetter

Geometry, difficulty level 3. Two people are arguing about whether three sections of fencing 6, 8, and 10 feet long will enclose more or less area than three sections 6, 8, and 12 feet long.
... more>>

Building a Vaulted Ceiling
 Annie Fetter

Geometry, difficulty level 4. An eyebrow window is a 15inchdeep slice off an 8.5foot circle. How wide will the base of the window be?
... more>>

Building Bookshelves
 Steve Weimar

Geometry, difficulty level 2. Merriem and Emmanuel's new bookshelf hits the ceiling when they try to
put it in place. Explain why, and help them figure out how much to
cut off the bottom so that it will fit.
... more>>

Calculating String Length
 Annie Fetter

Geometry, difficulty level 2. Given the archery targets of two competitors, figure out which one has the shortest 'string length'.
... more>>

Checking a Building Foundation for Square
 Annie Fetter

Geometry, difficulty level 2. How can you determine whether a rectangle is square when all you have is string and a tape measure?
... more>>

A Circle Inscribed in an Isosceles Triangle
 Ilmar Vitsut

Geometry, difficulty level 4. Find the radius of a circle inscribed in an isosceles triangle with sides 12, 12, and 8.
... more>>

Diagonal Dilemma
 Karen Cohen

Geometry, difficulty level 1. What is the area of the square ABCD that has a diagonal of length 12 cm?
... more>>

The Dimensions of a Hunting Platform
 Annie Fetter

Geometry, difficulty level 2. I have a picture of a hunting platform, and I need help figuring out some of the dimensions.
... more>>

Drawing a Kite Plan
 Annie Fetter

Geometry, difficulty level 3. Given a picture of a kite plan, figure out the radius of the circle that creates the curved section.
... more>>

Dueling Spheres
 Annie Fetter

Geometry, difficulty level 2. You're given two spheres with centers 14 cm apart. The circumference
of the circle where they intersect is 24 pi cm, and the radius of one
sphere is 13 cm. What's the radius of the other sphere?
... more>>

Examining an Octagon
 Annie Fetter

Geometry, difficulty level 3. Describe the rectangles that you would need to put on the sides of a square in order to be able to end up with a regular octagon when you connect the outside vertices of the figure.
... more>>

Folding a Rectangle
 Annie Fetter

Geometry, difficulty level 3. Fold a 6inch by 8inch piece of paper so that the opposite corners
meet. How long is the fold?
... more>>

Geoboard Gymnastics
 Annie Fetter

Geometry, difficulty level 3. Find squares with different areas on a geoboard.
... more>>

An Inscribed Circle
 Annie Fetter

Geometry, difficulty level 2. A circle is inscribed in a triangle with sides of 6, 6, and 4. What's the radius of the circle?
... more>>

Jill and Jimmy  Soap Bubbles
 Annie Fetter

Geometry, difficulty level 3. You're given two spheres with centers 14 cm apart. If the two spheres intersect in a circle whose radius is 12 cm, and the radius of one sphere is 13 cm, what's the radius of the other sphere?
... more>>

Leaning Ladder
 Annie Fetter

Geometry, difficulty level 3. A ladder is leaning against a wall. Explore what happens to the top
of the ladder when the bottom is moved closer or further from the wall.
... more>>

The Length of a Belt
 Annie Fetter

Geometry, difficulty level 3. The diameters of two wheels are 18 cm and 12 cm and their centers are 30 cm apart. A belt goes around them and crosses 18 cm from the center of the big wheel. How long is the belt? How do you know you're right?
... more>>

Mad About Medians?
 Annie Fetter

Geometry, difficulty level 3. The medians drawn from the vertices of the acute angles of a right triangle are 5 and sq root of 40. What is the length of the hypotenuse?
... more>>

More Tangents!
 Annie Fetter

Geometry, difficulty level 3. Given a square ABCD inscribed in a unit circle. AB is extended to P, where PC is tangent to the circle. Find the length of PD.
... more>>

Olympic Softball
 Annie Fetter

Geometry, difficulty level 2. Should the pitcher's rubber in Olympic softball be behind the line drawn from third base to first base, or in front of it?
... more>>

A Picturesque Pythgorean Proof
 Annie Fetter

Geometry, difficulty level 3. Write a proof to go with this picture.
... more>>

Pythagorean Circles
 Lillian Ray

Geometry, difficulty level 3. Figure out the relationship between the area of three regions in this
construction
of "Pythagorean circles".
... more>>

Sighting in a Rifle
 Annie Fetter

Geometry, difficulty level 3. Analyze the given target of four shots, and tell me how far to move the scope on my rifle so that the group of shots will be centered around the bull'seye.
... more>>

Slicing a Cube
 Annie Fetter

Geometry, difficulty level 3. Given a cube with a surface area of 54 cm^2, find the perimeter of the square cross section and that of the largest rectangular cross section of that cube.
... more>>

The Speed Skating Oval
 Annie Fetter

Geometry, difficulty level 2. Find the radii of the turns of a 400 meter speed skating oval.
... more>>

Steam Up!
 Annie Fetter

Geometry, difficulty level 3. Calculating the grade of an incline and the length of a train.
... more>>

Taking Pictures of the Earth
 Annie Fetter

Geometry, difficulty level 4. If you wanted to go out into space and take a picture of the earth, how far from the earth would you have to go to make sure that you got 1/4 of the equator in your picture? How about 1/3?
... more>>

A Tangent Square and Circle
 Reza Kassai

Geometry, difficulty level 3. The square and circle below are tangent at one point and meet at four
other points, as shown in the picture. If the side of the square is 8
units long, what's the radius of the circle?
... more>>

Two Houses and a Carrier Pigeon
 Annie Fetter

Geometry, difficulty level 3. One house is 300 yards from a highway. 1200 yards down the highway is the driveway to another house, 600 yards from the highway, on the same side of the road. What is the shortest distance between the two houses if you go to the highway in between?
... more>>

What Do We Know About Tangents?
 Annie Fetter

Geometry, difficulty level 2. Given two externally tangent congruent circles, construct a segment from the center of one circle tangent to the other circle. If the length of this segment is 10, what's the radius of the circles?
... more>>

Who Wants to be an Archery Champion?
 Annie Fetter

Geometry, difficulty level 2. To determine the winner, calculate the "string length" for each of two archery rounds.
... more>>

World Series Snafu?
 Annie Fetter

Geometry, difficulty level 2. Determine what's wrong with the "official" dimensions of "home plate" in baseball.
... more>>
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