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Dr. Math FAQ:
Browse High School Triangles and Other Polygons
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Area of an irregular shape.
Pythagorean theorem proofs.
- Proving Lines Congruent [03/29/2002]
Prove line AL is congruent to line CM.
- Proving Quadrilateral is a Parallelogram [11/30/2001]
We are having a problem with the idea of a quadrilateral having one pair
of opposite sides congruent and one pair of opposite angles congruent.
- Proving Quadrilateral Is a Parallelogram, Redux [04/04/2012]
A geometry teacher wonders if his student has proven that a quadrilateral with one
pair of congruent angles and one set of congruent angles is a parallelogram. By
following the steps from another Dr. Math conversation cited by the teacher, Doctor
Peterson illustrates the proof's hidden assumption with a counter-example.
- Proving the Diagonals of a Rectangle Congruent [12/6/1995]
How would you prove that the diagonals of a rectangle are congruent?
- Proving the Existence of the Centroid [11/16/1999]
How can I prove that the centroid of any arbitrary triangle exists?
- Proving the Pythagorean Theorem: A Traditional and a Modern Approach [01/27/1998]
Can you please explain how I can prove the Pythagorean theorem?
- Proving the Pythagorean Theorem in Two Steps [1/28/1996]
I was trying to prove the Pythagorean theorem for a friend of mine, and
eventually we figured it out. But then he said that he had heard of a
"proof in only six easy steps." Do you know anything of this?
- Proving the Pythagorean Theorem using Congruent Squares [12/5/1995]
A friend of mine is irked because of constant use of the Pythagorean
theorem, which he has not seen proven.
- Proving Trapezoid Congruency [12/19/2001]
Prove that the sides of a trapezoid are congruent if the diagonals of the
trapezoid are congruent.
- Ptolemy's Theorem [09/07/1997]
Can you give me a reference for the proof for Ptolemy's Theorem?
- A 'Pyramiddle' Tent Problem [07/12/1999]
Figure out an equation that yields d when values for h and r are
- The Pythagorean Theorem: A Modern Proof [04/14/1997]
I know that the Pythagorean Theorem works and I can show how it works,
but why does it work?
- Pythagorean Theorem and Cubes [02/14/1998]
In a cube if a diagonal is drawn from the front top corner to the back
bottom corner, how long must each side be using the Pythagorean Theorem?
- Pythagorean Theorem - Euclid's Proof [12/27/1998]
A detailed explanation of a specific proof.
- Pythagorean Theorem, Fermat's Last Theorem [5/16/1996]
Can the Pythagorean theorem be done with 3 different numbers?
- Pythagorean Theorem in Three Dimensions [05/18/2001]
Given a tetrahedron with a trirectangular vertex S. Let A, B, and C be
the areas of the three faces that meet at S, and D be the area of the
face opposite S. Prove that D^2 = A^2 + B^2 + C^2.
- Pythagorean Theorem Legend [01/31/2009]
Is there a Sioux Indian legend that refers to the Pythagorean Theorem?
- Pythagorean Theorem Proof by Brodie Explained [03/15/2000]
Could you give me a step-by-step explanation of Dr. Scott Brodie's proof
of the Pythagorean theorem given at the cut-the-knot website?
- Pythagorean Theorem Proof: Four Right Triangles [10/7/1996]
I don't understand the Pythagorean Theorem.
- Pythagorean Theorem Proof (Thabit ibn Qurra) [03/28/2002]
Proving a series of congruent triangles.
- Pythagorean Theorem vs. Square's Diagonal [03/19/2003]
Imagine that you want to get from one corner of a right triangle to
the other via the hypotenuse, but can only make moves perpendicular to
the other two sides.
- Pythagorean Triple [8/28/1996]
What is the formula for finding the three lengths in a Pythagorean triple
where the shortest side is even?
- Pythagorean Triples [10/07/1997]
What is a Pythagorean triple?
- Pythagorean Triples [04/14/1997]
Why can't all the numbers in a Pythagorean triple be prime?
- Pythagorean Triples [5/18/1995]
How can the relation between Pythagorean triples be expressed as a
- Quadrilateral Area [04/29/2002]
Given the (x,y) coordinates of four points, is there a simple formula
to compute the area of the quadrilateral with corners on these 4
- Quadrilateral Area Given Side Lengths [8/24/1996]
I need to find the area of a 4-sided figure, given its side lengths.
- Quadrilateral Classification: Definition of a Trapezoid [01/15/1997]
What is the correct definition of a trapezoid, and why?
- Quadrilateral Patterns [03/06/1997]
What is the formula that gives the number of quadrilaterals within a
square grid when you increase the square grid by one unit on each side?
- Quadrilateral Problem [5/8/1995]
If ABCD is a convex quadrilateral and M, N, P, Q are points on AB, BC,
CD, DA respectively, prove that...
- Quadrilaterals and Diagonals [1/18/1995]
If the diagonals of a quad are congruent, must the quadrilateral be a rectangle or
an isosceles trapezoid?
- Quadrilaterals and Inscribed Circle [05/06/1999]
From ten sticks of lengths 1,2,3,....,10 four are selected to form the
sides of a quadrilateral...
- Radius of a Circle Inscribed in a Triangle [06/02/1999]
What is the radius of an inscribed circle of a triangle with sides 3, 4,
- Radius of Circumscribed Circle [05/11/2001]
Where can I find a derivation of R = abc/4K?
- Rail Bend in Hot Weather [10/13/2002]
A 20-ft piece of rail expands 1 in. in length during a hot spell. If
there are no expansion gaps, how high off the ground will the rail
- Railroad Track Expansion [04/07/2003]
A continuous straight railroad track of one mile is permanently tied
down at both ends. As the day heats up, the coefficient of expansion
of steel causes the rail to expand so that the length is now 5281
feet. Assuming that the track expands upward, what maximum vertical
distance from the horizontal will the track rise at the highest point?
- Ratio and proportion [12/5/1994]
I need extra help in ratio and proportion.
- Ratio of Areas of Triangle and Parallelogram [7/17/1996]
Given a parallelogram in a triangle... compute the area of the triangle
divided by the area of the parallelogram.
- Ratio of Sides and Ratio of Areas [02/11/1999]
If the sides of a triangle are in the ratio of 1:7, what is the ratio of
their areas? What about for other shapes?
- Ratios and Geometry [10/29/1998]
An airplane flying at 33,000 feet has a visibility of 100 miles. What
percent of the total land area to the horizon is visible?