# Tag Archives: geometry

## Tangent Lines

Suppose to co-planar circles and have no points in common. Determine how many lines, if any, are tangent to both and ? Source: NCTM Mathematics Teacher, October 2006 Solution When one circle is inside the other, there are no tangent … Continue reading

## Three Intersecting Circles

Circle has a radius of and its center is at . Circle has a radius of and its center at . What is the length of the radius of a third circle (Circle ) that passes through the center of … Continue reading

## Rectangle Inside a Square

An isosceles right triangle is removed from each corner of a square piece of paper so that a rectangle remains. What is the length of a diagonal of the rectangle if the sum of the areas of the removed pieces … Continue reading

## Equiangular Hexagon

An equiangular hexagon with consecutive side lengths , and can be inscribed in an equilateral triangle with side length . This same equiangular hexagon can also be inscribed in a second (distinct, noncongruent) equilateral triangle with side length . What … Continue reading

## Vertices of a Square

If and are the coordinates of two opposite vertices of a square, what are the coordinates of the other two vertices? Source: NCTM Mathematics Teacher, October 2006 Solution Let and be the two given opposite vertices of the square. The … Continue reading

## Least Side Length of Right Triangle

If the lengths of two sides of a right triangle are and , what is the least possible length of the third side? Source: NCTM Mathematics Teacher, October 2006 Solution If the legs of the right triangle measure and units, … Continue reading

## Convex Polygon

What is the maximum number of acute angles that any convex polygon can have? Source: NCTM Mathematics Teacher, January 2006 Solution The Polygon Exterior Angles Theorem states that the sum of the measures of the exterior angles of a convex … Continue reading

## Lattice Points

A point with integral coordinates is called a lattice point. For example, is a lattice point, whereas is not. How many lattice points lie on the graph of ? Source: NCTM Mathematics Teacher, January 2006 Solution The graph of is … Continue reading

## Corners of a Cube

All corners were sliced off a cube whose volume was cubic centimeters. Triangular faces are located where the cube’s vertices were. Each triangle is equilateral with -centimeter sides. Determine the total numbers of edges on the new solid. Source: NCTM … Continue reading

## Circular Garden

Ebru’s backyard is a square with side length of meters. Her backyard has a circular garden that extends to each side of the yard. In the center of the garden is a square patch of asparagus so big that each … Continue reading