Section 1
Cartesian Coordinates
find real-number values, if there are any, for which the equation is true. If no such real values exist, so state.$(x+3,5)=(-1,9+x)$
find real-number values, if there are any, for which the equation is true. If no such real values exist, so state.$(x-4,2)=(3, x-5)$
find real-number values, if there are any, for which the equation is true. If no such real values exist, so state.$(2 x-7, x+2)=(-5,3)$
find real-number values, if there are any, for which the equation is true. If no such real values exist, so state. $(3 x+2,2 x-3)=(8,1)$
find real-number values, if there are any, for which the equation is true. If no such real values exist, so state.$(x-2 y, 2 x+y)=(-1,3)$
find real-number values, if there are any, for which the equation is true. If no such real values exist, so state.$(2 x+3 y, x+4 y)=(3,-1)$
find real-number values, if there are any, for which the equation is true. If no such real values exist, so state.$\left(x^2-2 x, x^2-x\right)=(3,6)$
find real-number values, if there are any, for which the equation is true. If no such real values exist, so state.$\left(x^2+2 x, 2 x^2+3 x\right)=(-1,-1)$
find the distance between the given points $\mathbf{S}$ and $\mathbf{T}$. Express results in simplest radical form.$\mathbf{S}(1,3), \mathbf{T}(-2,6)$
find the distance between the given points $\mathbf{S}$ and $\mathbf{T}$. Express results in simplest radical form.$S(1,-6), T(6,6)$
find the distance between the given points $\mathbf{S}$ and $\mathbf{T}$. Express results in simplest radical form.$S(\sqrt{2}, \sqrt{2}), T(-\sqrt{2},-\sqrt{2})$
find the distance between the given points $\mathbf{S}$ and $\mathbf{T}$. Express results in simplest radical form.$\mathbf{S}(\sqrt{3},-\sqrt{3}), \mathbf{T}(-\sqrt{3}, \sqrt{3})$
find the distance between the given points $\mathbf{S}$ and $\mathbf{T}$. Express results in simplest radical form. $S(4, \sqrt{3}), T(2,-1)$
find the distance between the given points $\mathbf{S}$ and $\mathbf{T}$. Express results in simplest radical form. $\mathbf{S}(\sqrt{2},-\sqrt{3}), \mathbf{T}(1,2)$
Show that the triangle with vertices at the points $\mathbf{R}(0,1), \mathbf{S}(8,-7)$, and $\mathbf{T}(1,-6)$ is isosceles.
Show that the points $\mathbf{R}(-4,4), \mathbf{S}(-2,-4)$, and $\mathbf{T}(6,-2)$ are the vertices of an isosceles triangle.
Show that the point $\mathbf{Q}(1,-2)$ is equidistant from the points $\mathbf{R}(-11,3)$, $\mathbf{S}(6,10)$, and $\mathbf{T}(1,11)$.
Show that the point $\mathbf{Q}(2,-3)$ is equidistant from the points $\mathbf{R}(6,0)$, $\mathbf{S}(-2,-6)$, and $\mathbf{T}(-1,1)$
The points $\mathbf{Q}(1,1), \mathbf{R}(2,5), \mathbf{S}(6,8)$, and $\mathbf{T}(5,4)$ are the vertices of a quadrilateral. Show that the opposite sides of the quadrilateral are equal in length.
Are the opposite sides of the quadrilateral whose vertices are the points $\mathbf{Q}(-2,3), \mathbf{R}(5,2), \mathbf{S}(7,-4)$, and $\mathbf{T}(0,-2)$ equal in length?
Use the distance formula to show that the points $\mathbf{R}(-2,-5), \mathbf{S}(1,-1)$, and $\mathbf{T}(4,3)$ lie on the same line.
Show that the points $\mathbf{R}(-3,3), \mathbf{S}(2,1)$, and $\mathbf{T}(7,-1)$ lie on the same line.
Show that $\mathbf{R}(1,5)$ is the midpoint of the line segment with endpoints $\mathbf{S}(-2,3)$ and $\mathbf{T}(4,7)$.
Show that $\mathbf{M}\left(\frac{a+c}{2}, \frac{b+d}{2}\right)$ is the midpoint of the line segment with endpoints $\mathbf{S}(a, b)$ and $\mathbf{T}(c, d)$.
Find the midpoint of the line segment with endpoints $\mathbf{S}(-2,9)$ and $\mathbf{T}(8,-1)$.
Find the midpoint of the line segment with endpoints $\mathbf{S}(-3,5)$ and $\mathbf{T}(3,2)$.
Show that for points $\mathbf{A}\left(x_1, 0\right)$ and $\mathbf{B}\left(x_2, 0\right), d(\mathbf{A}, \mathbf{B})=\left|x_2-x_1\right|$.
Show that for points $\mathbf{C}\left(0, y_1\right)$ and $\mathbf{D}\left(0, y_2\right), d(\mathbf{C}, \mathbf{D})=\left|y_2-y_1\right|$.