Ask Dr. Math

High School Archive

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.
    Classifying quadrilaterals.
    Heron's formula.
    Polygon diagonals.
    Pythagorean theorem proofs.
    Triangle congruence.

---

Triangle Inequality Theorem [03/09/2001]

The lengths of the sides of a non-isosceles triangle, in size order, are 5, x, and 15. What are all possible integral values of x?

Triangle: Longest Side Opposite Greatest Angle [10/23/1999]

Prove that in any triangle, the greatest side is opposite the greatest angle.

Triangle Midpoints and Vertices [02/23/2003]

How do you figure out the vertices of a triangle algebraically by using its three midpoints?

Triangle Perimeter [07/20/1997]

How many triangles have sides whose lengths total 15 units?

Triangle Proof [2/18/1995]

Maybe if two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. I'm not sure how to say this in a proof.

Triangle Proof: Parallel Sides and Proportionality [07/27/2004]

How do I prove that a line which cuts two sides of a triangle proportionately is parallel to the third side?

Triangle Proof: r + r1 + r2 = CD [04/20/2001]

Let CD be an altitude of triangle ABC, and assume that angle C = 90 degrees. Let r1 and r2 be the inradii of triangle CAD and triangle CBD, respectively, and show that r+r1+r2=CD, where r is the inradius of triangle ABC.

Triangle Proofs in General [11/19/2001]

Mapping out a general method for proceeding with proofs.

Triangle Proof with Contradiction [02/21/2004]

Let D, E lie internally on side BC of triangle ABC and consider the following conditions: 1) angle BAD = angle DAE = angle EAC 2) |BD| = |DE| = |EC| Prove that, whatever the shape of triangle ABC, 1) and 2) cannot both be true, that is, if either is true, then the other is false.

Triangle Proportions: A Diagram [05/06/2003]

A cone has a circular base radius 1, and vertex of height 3 directly above the center of the circle. A cube has four vertices in the base and four on the sloping sides. What is the length of a side of the cube?

Triangles: Angle Sums [05/15/2002]

Can you draw a triangle in which the sum of any two angles - no matter which two you pick - is always less than 120 degrees?

Triangles in a Polygon [06/14/1997]

A regular 18-sided polygon is inscribed in a circle and triangles are formed by joining any three of the eighteen vertices. How many obtuse triangles are there?

Triangle's Medians Make Smaller Triangles with Equal Area [04/15/1999]

Proving that the six triangles constructed from the three medians of any triangle have the same area.

Triangle Vertices But Not Sides [02/22/2003]

If P is a regular n-gon, what is the number of triangles whose vertices are the vertices of P but whose sides are NOT the sides of P?

A Triangle with Three Right Angles [12/01/1999]

How can you make a triangle with three right angles?

Triangular Garden [03/18/1997]

Find the length of a fence that runs from the right angle to the hypotenuse and separates the garden into two parts of equal perimeter.

Trisected Hypotenuse of a Triangle [12/20/1998]

In right triangle ABC, with C as the right angle... what is the length of AB (the hypotenuse)?

Trisecting an Angle and the Opposite Side in a Triangle [09/03/2008]

Prove that it is impossible to have a triangle in which the trisectors of an angle also trisect the opposite side.

Truncating a Square to Get an Octagon [10/13/2003]

I want to make an octagon by cutting the corners off of a square. Where do I make the cuts?

Twenty Quadrilaterals from Nine Dots [04/04/1999]

How can you get 20 quadrilaterals from 9 dots?

Two-Column Proof About Kites [11/09/1999]

Can you help me understand a proof about perpendicular lines and congruent triangles in a kite?

Two Column Proof of a Theorem [08/12/1998]

Write a two-column proof and give numbered statements with reasons....

Two Questions on Geometric Harmonics [11/24/2005]

Two circles intersect each other at B and C. Their common tangent touches them at P and Q. A circle is drawn through B and C cutting PQ at L and M. Prove that {PQ:LM} is harmonic.

Two-Sided Polygon? [12/01/2003]

My 5th grade math teacher said that we had to draw a polygon using two straight lines. Is this possible?

Understanding Bearings in Directional Problems [01/14/2004]

A boat sails 10km from a harbor H on a bearing of S30 degree E. It then sails 15 km on a bearing of N20 degree E. How far is the boat from H? What is the bearing from H?

Understanding Rectangle Area and Perimeter [11/08/2002]

True or false: if the perimeter of a rectangle increases, the rectangle's area always also increases.

Uniquely Determining a Polygon [02/05/2001]

Is it true that if you know the side order, side lengths, and area of a polygon, as well as whether each of its angles is obtuse or acute, you have uniquely determined it?

Using Midpoints to Determine Vertices [09/04/2002]

The midpoints of the sides of a triangle have coordinates G(3,1), H (-1,2) and J (1,-3). Determine the coordinates of the vertices of the triangle.

Using the Incenter [05/06/2003]

I need to construct a triangle to fit inside a triangle.

Vectors of Parallelograms and Octagons [07/28/1998]

ABCDEFGH is a regular octagon and AB = p and BC = q. Express AH in terms of p and q...

Venn Diagram to Classify Quadrilaterals [01/02/2003]

I am looking for a Venn diagram that will accurately display the relation among trapezoids, parallelograms, kites, rhombi, rectangles, and squares.

What Does a Myriagon Look Like? [10/30/2002]

I am looking for a picture of a myriagon.

What does Angle ABC Equal? [3/5/1995]

A triangle, ABC, is obtuse angled at C. The bisectors of the exterior angles at A and B meet BC and AC produced at D and E respectively. If AB=AD=BE, then what does angle ABC equal?

What is an N-gon? [06/01/1998]

Can you explain the statement: "In an N-gon, n-3 diagonals can be drawn from one vertex"?

What is Length in a Rectangle? [05/31/1999]

Is the length of a rectangle the longest side, whether vertical or horizontal?

What is Menelaus' Theorem? [11/15/1998]

Proof of Menelaus' Theorem, and discussion of its converse and Desargues' Theorem.

What is the Area Not Shared by the Circles? [3/3/1995]

Two circles intersect such that their centers and their points of intersection form a square with each side equal to 3. What is the total area of the sections of the square that are not shared by both circles?

What Is the Length of PR? [01/01/2003]

In a circle of radius 6, a triangle PQR is drawn having QR=8 and PQ=10.

Why are Manhole Covers Round? [05/09/2000]

Why are most manhole covers round? Why aren't manhole covers on the streets squares or rectangles?

Why a Square Maximizes Area/Perimeter [07/24/2002]

Is it possible to make a rectangle with a perimeter of 16 feet and an area greater than 16 square feet?

Page: [<prev]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 [next>]