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The Euclidean metric (used in the Euclidean plane) is [math]d\left(\left(x_1,y_1\right),\left(x_2,y_2\right)\right)=\sqrt{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}[/math]. The distinct between two points is exactly the length of the line segment that connects them.[br][br]The Taxicab metric on the Cartesian plane is [math]d\left(\left(x_1,y_1\right),\left(x_2,y_2\right)\right)=\left|x_1-x_2\right|+\left|y_1-y_2\right|[/math]. This distance can be understood visually as the length of a shortest path taken between the two points that only moves horizontally or vertically.
Find the Euclidean distance between [math]\left(-5,11\right)[/math] and [math]\left(3,7\right)[/math].
Find the distance between [math]\left(-5,11\right)[/math] and [math]\left(3,7\right)[/math] in the taxicab metric.
In the graph below, move the points [math]A,B,C[/math] to form your own triangle and compute the perimeter of [math]\Delta ABC[/math] using the taxicab metric.
The set of points in our model of taxicab geometry is the usual Cartesian plane. The set of lines in this model is the usual set of lines in the plane. Let [math]P=\left(x_1,mx_1+b\right)[/math] and [math]Q=\left(x_2.mx_2+b\right)[/math] be two points on the non-vertical line [math]y=mx+b[/math]. Find [math]d\left(P,Q\right)[/math]. In particular, if you write [math]d\left(P,Q\right)=k\left|x_1-x_2\right|[/math], what is [math]k[/math]?
The ruler postulate says we must be able to assign real number coordinates to lines in a way that agrees with the given metric of the geometry. If [math]k[/math] is the value you just found, then you can assign coordinates to the line [math]y=mx+b[/math] by giving a point [math]P\left(x,y\right)[/math] the coordinate [math]f\left(P\right)=kx[/math]
Angle measure in taxicab geometry is the same as euclidean geometry. In particular, an angle in taxicab geometry is a right angle if and only if it is a right angle in Euclidean geometry.
In the graph above, find the measure of [math]\overline{AC}[/math],[math]\overline{BC}[/math], and angle [math]\angle ACB[/math]
In the graph above, find the measure of [math]\overline{CD}[/math],[math]\overline{BD}[/math], and angle [math]\angle BDC[/math]
Your computations should show that under a certain correspondence of vertices, these two triangles satisfy the SAS criterion. Why are the two triangles not congruent in taxicab geometry?
The circle centered at [math]\left(a,b\right)[/math] with radius [math]r[/math] is the set [math]\left\{\left(x,y\right)\mid d\left(\left(x,y\right),\left(a,b\right)\right)=r\right\}[/math]. In other words, you want the set of all [math]\left(x,y\right)[/math] satisfying [math]\left|x-a\right|+\left|y-b\right|=r[/math]. Fix [math]a,b,r[/math] in the plot below. What does a circle look like in taxicab geometry?
Let [math]A\left(a,b\right)[/math] and [math]B\left(c,d\right)[/math] be distinct points. The perpendicular bisector of [math]\overline{AB}[/math] is the locus of points equidistant from [math]A[/math] and [math]B[/math], that is, the set of points [math]\left(x,y\right)[/math] satisfying [math]d\left(\left(x,y\right),\left(a,b\right)\right)=d\left(\left(x,y\right),\left(c,d\right)\right)[/math]. Using the taxicab metric for [math]d[/math], pick a pair of points in the graph below and plot the perpendicular bisector of [math]\overline{AB}[/math]. Is the perpendicular bisector a line?
An ellipse with foci [math]A\left(a,b\right)[/math] and [math]B\left(c,d\right)[/math] is the set of points [math]X\left(x,y\right)[/math] such that the sum of distances from [math]A[/math] to [math]X[/math] and [math]B[/math] to [math]X[/math] is a fixed distance. This means you want the set of [math]\left(x,y\right)[/math] such that there is a fixed number [math]r>0[/math] such that [math]d\left(\left(a,b\right),\left(x,y\right)\right)+d\left(\left(c,d\right),\left(x,y\right)\right)=r[/math].[br][br]In the graph below, fix your points [math]A,B[/math] and choose a number [math]r>0[/math]. Plot the equation of this ellipse using the taxicab metric for [math]d[/math]. What does an ellipse look like in taxicab geometry?