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- Line and Unit Circle; Pythagorean Triples [04/16/2001]
If (X,Y) is a point in the 1st quadrant on the unit circle and m is the
slope of the line passing through (X,Y) and the point (0,-1), how can I
express the coordinates (X,Y) in terms of m? Can this be used to generate
- Line Dividing a Plane [05/11/2001]
Given a square (graphed on the Cartesian coordinate system) and a point
in the square, draw a line through the point that will divide the square
into two regions: one the smallest area possible, the other the largest
- Locus and Equations of Lines [01/10/1999]
Describe the locus of points that are 3 units from the line x = -1...
- Locus of the Midpoint of a Chord [03/13/1998]
Show that the locus of the midpoint of the chord is a hyperbola.
- Midpoint Formula and Trisection Points [09/05/2003]
Show that ((1/3)(2x_1+ x_2), (1/3)(2y_1+y_2)) is one of the points of
trisection of the line segment joining (x_1,y_1) and (x_2,y_2).
- Midpoint Formula for any Fraction [01/26/1999]
Is there a formula similar to the midpoint formula to find "any
fractional part" of the line from P1 to P2?
- Midpoint of a Straight Line Segment [5/7/1996]
What is the midpoint of this: (-3,4) (5,-4)? Use the distance formula.
- Minimum Distance from a Point to a Line [7/10/1995]
Find all the values of b such that the minimum distance from the point
(2,0) to the line y = 4/3x+b is 5.
- Nonconvex Polygon Angle Measure [02/03/1999]
What is the formula to find the interior angle measurements of a
- Ordered Pairs vs. Coordinates [9/9/1996]
What is the difference between an ordered pair and coordinates?
- Order of Quadrant Numbers [10/21/2002]
Why are quadrants numbered that way?
- Overlapping Polygons [06/10/2003]
Given all the coordinates for two different polygons in an x,y plane,
how can I determine whether the polygons overlap?
- Pick's Formula [05/22/1997]
What is Pick's Formula?
- Pick's Theorem, Lattice Points, and Area [08/27/1998]
What is a lattice point, and how does it relate to the area of a
triangle, rectangle, and a circle?
- Point Inside or Outside Triangle? [08/02/2001]
I have coordinates of three vertices of the triangle and coordinates of
- Point on an Ellipse [05/16/1997]
Given an ellipse and an arbitrary angle theta from either axis, how do
you find the coordinates of the intersection of the ellipse and a vector
formed by angle theta?
- Point on a Plane [06/13/1997]
If I know the coordinates of three points that form a plane and the first
two coordinates of another point on that plane, how do I find the third
coordinate of that point?
- Point Reflected in a Plane [6/7/1995]
How do I find the coordinates of point p' which is the reflection of
point p in the plane E?
- Programming the Distance Formula [03/26/2001]
I want to write a computer program to find the point on a line that's a
distance X from a given point not on the line. Can you help?
- Proof of Perpendicularity [10/23/1999]
How can you prove that two lines (neither vertical) are perpendicular if
and only if the product of the gradients is equal to -1?
- A Proof using Analytic Geometry [02/24/1999]
Prove that, if p is a point inside the ellipse, there is one and only one
chord QP bisected at P.
- Proof Using Coordinate Geometry [06/08/2001]
In equilateral triangle ABC, a segment is drawn from point A to the side
BC at a point (D)... prove that angle BFC (or EFC) is a right angle.
- Proving Diagonals Perpendicular [07/28/2003]
Given the points a(-4,1), b(2,3), c (4,9) and d (-2,7), show that
quadrilateral abcd is a parallelogram with perpendicular diagonals.
- Quadrant [03/25/2001]
What is a quadrant?
- 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
- Radius and Center of a Circle from 3 Points [07/23/1999]
Given the coordinates of three points on a circle, how can you find the
center and radius?
- Radius, Center of Circle Given 3 Points [8/28/1996]
What is an easy formula to calculate the center point and radius of a
circle given three points on the circumference?
- A Right Triangle of Points [01/14/1999]
Determine the values of x that would make the points (x,0), (-2,1), and
(3,4) the vertices of a right triangle.
- Similar Triangles [1/22/1996]
For triangle ABC whose vertices are A(6,3),B(1,5),C(-1,4), what are the
vertices of a similar triangle whose perimeter is 5 times that of
- A Slope Problem Involving a Billiards Table [10/19/2004]
Suppose that a square billard table has corners at (0,0),(1,0),(1,1),
(0,1). A ball leaves the origin along a line with slope s. If the ball
reaches a corner, it stops or falls in. Whenever it hits the side of
the table not at a corner, it continues to travel on the table, but
the slope on the line is multiplied by -1. Does the ball reach a corner?
- Solve for Radius [09/09/1997]
Given a circle through three points, what is the equation for the
intersection of the perpendicular bisectors?
- Stretching and Shearing away from Coordinate Axes [08/27/2012]
A student struggles to calculate matrices that correspond to linear transformations that
occur off of the x- or y-axes. Doctor Schwa formalizes what the student has already done, then introduces a second approach and the concept of conjugations.
- Tangent Common to Two Ellipses [10/24/2002]
I have two general ellipses in space and I want to find the equation
of the tangent common to these ellipses.
- Theta [04/14/1997]
What is Theta? Does it have a constant value?
- Triangle Circumcenter's Coordinates [08/15/2012]
Seeking the circumcenter of a triangle drawn on the coordinate plane, a student
determines the midpoints and slopes of all three legs -- but the equation for only one
of its perpendicular bisectors. Doctor Schwa offers a small hint that she uses to finish
off the proof.
- 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 Cartesian Plane [01/19/1997]
Draw a rectangle that is Sqrt(2) by 1 with corners at (0,0), (1,0),
(1,Sqrt(2)), and (0,Sqrt(2)).
- Vertex of a Parabola [04/16/2002]
Finding the y-coordinate of the vertex of a parabola.
- A Way to Remember What 'm' and 'b' Mean in Slope-Intercept Form [02/01/2004]
An interesting memory trick to help you remember how to graph a linear
equation in slope-intercept form.
- What is a point? [8/26/1996]
Define a point, please.