Proving Theorems with Coordinate Geometry - Geometry
Card 1 of 30
Identify whether $\overline{CD}$ is vertical if $C(2,-1)$ and $D(2,9)$.
Identify whether $\overline{CD}$ is vertical if $C(2,-1)$ and $D(2,9)$.
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Yes, since $x_1=x_2$. Same $x$-coordinate indicates vertical segment.
Yes, since $x_1=x_2$. Same $x$-coordinate indicates vertical segment.
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Identify the distance between $(0,0)$ and $(3,4)$.
Identify the distance between $(0,0)$ and $(3,4)$.
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$5$. $d = \sqrt{3^2+4^2} = \sqrt{25} = 5$
$5$. $d = \sqrt{3^2+4^2} = \sqrt{25} = 5$
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What slope relationship shows $\overline{AB}$ is perpendicular to $\overline{BC}$?
What slope relationship shows $\overline{AB}$ is perpendicular to $\overline{BC}$?
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$m_{AB}m_{BC}=-1$ (or one vertical, one horizontal). Product of slopes equals $-1$ for perpendicular segments.
$m_{AB}m_{BC}=-1$ (or one vertical, one horizontal). Product of slopes equals $-1$ for perpendicular segments.
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Identify the slope of the line through $(2,3)$ and $(6,11)$.
Identify the slope of the line through $(2,3)$ and $(6,11)$.
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$m=2$. $m = \frac{11-3}{6-2} = \frac{8}{4} = 2$
$m=2$. $m = \frac{11-3}{6-2} = \frac{8}{4} = 2$
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Identify the midpoint of $( -2,5)$ and $(4,1)$.
Identify the midpoint of $( -2,5)$ and $(4,1)$.
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$\left(1,3\right)$. $\left(\frac{-2+4}{2}, \frac{5+1}{2}\right) = (1,3)$
$\left(1,3\right)$. $\left(\frac{-2+4}{2}, \frac{5+1}{2}\right) = (1,3)$
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Identify whether lines with slopes $\frac{3}{4}$ and $\frac{3}{4}$ are parallel.
Identify whether lines with slopes $\frac{3}{4}$ and $\frac{3}{4}$ are parallel.
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Yes, they are parallel. Equal slopes confirm parallel lines.
Yes, they are parallel. Equal slopes confirm parallel lines.
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Identify whether lines with slopes $2$ and $-\frac{1}{2}$ are perpendicular.
Identify whether lines with slopes $2$ and $-\frac{1}{2}$ are perpendicular.
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Yes, they are perpendicular. Product $2 \cdot (-\frac{1}{2}) = -1$ confirms perpendicular.
Yes, they are perpendicular. Product $2 \cdot (-\frac{1}{2}) = -1$ confirms perpendicular.
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Identify whether lines with slopes $\frac{5}{2}$ and $\frac{2}{5}$ are perpendicular.
Identify whether lines with slopes $\frac{5}{2}$ and $\frac{2}{5}$ are perpendicular.
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No, since $\frac{5}{2}\cdot\frac{2}{5}=1$. Product equals $1$, not $-1$, so not perpendicular.
No, since $\frac{5}{2}\cdot\frac{2}{5}=1$. Product equals $1$, not $-1$, so not perpendicular.
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Identify whether $\overline{AB}$ is horizontal if $A(-3,7)$ and $B(5,7)$.
Identify whether $\overline{AB}$ is horizontal if $A(-3,7)$ and $B(5,7)$.
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Yes, since $y_1=y_2$. Same $y$-coordinate indicates horizontal segment.
Yes, since $y_1=y_2$. Same $y$-coordinate indicates horizontal segment.
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Identify whether $\overline{CD}$ is vertical if $C(2,-1)$ and $D(2,9)$.
Identify whether $\overline{CD}$ is vertical if $C(2,-1)$ and $D(2,9)$.
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Yes, since $x_1=x_2$. Same $x$-coordinate indicates vertical segment.
Yes, since $x_1=x_2$. Same $x$-coordinate indicates vertical segment.
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Identify the radius of the circle centered at $(0,0)$ passing through $(0,2)$.
Identify the radius of the circle centered at $(0,0)$ passing through $(0,2)$.
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$r=2$. Distance from origin to point $(0,2)$ is $2$.
$r=2$. Distance from origin to point $(0,2)$ is $2$.
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Which option is the slope of the line through $(1,-2)$ and $(5,6)$?
Which option is the slope of the line through $(1,-2)$ and $(5,6)$?
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$m=2$. $m = \frac{6-(-2)}{5-1} = \frac{8}{4} = 2$
$m=2$. $m = \frac{6-(-2)}{5-1} = \frac{8}{4} = 2$
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Identify whether $(1,\sqrt{3})$ lies on $x^2+y^2=4$.
Identify whether $(1,\sqrt{3})$ lies on $x^2+y^2=4$.
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Yes, since $1^2+(\sqrt{3})^2=4$. Substitution: $1 + 3 = 4$ satisfies equation.
Yes, since $1^2+(\sqrt{3})^2=4$. Substitution: $1 + 3 = 4$ satisfies equation.
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Identify whether $(2,1)$ lies on the circle $x^2+y^2=4$.
Identify whether $(2,1)$ lies on the circle $x^2+y^2=4$.
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No, since $2^2+1^2=5\neq^4$. Substitution: $4 + 1 = 5 \neq 4$.
No, since $2^2+1^2=5\neq^4$. Substitution: $4 + 1 = 5 \neq 4$.
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Identify the equation of the circle centered at $(3,-1)$ with radius $5$.
Identify the equation of the circle centered at $(3,-1)$ with radius $5$.
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$(x-3)^2+(y+1)^2=25$. Standard form with center $(3,-1)$ and radius $5$.
$(x-3)^2+(y+1)^2=25$. Standard form with center $(3,-1)$ and radius $5$.
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Identify whether $(6,3)$ lies on $(x-3)^2+(y+1)^2=25$.
Identify whether $(6,3)$ lies on $(x-3)^2+(y+1)^2=25$.
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Yes, since $(6-3)^2+(3+1)^2=25$. Substitution: $9 + 16 = 25$ satisfies equation.
Yes, since $(6-3)^2+(3+1)^2=25$. Substitution: $9 + 16 = 25$ satisfies equation.
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Identify $AB$ for $A(1,2)$ and $B(4,6)$.
Identify $AB$ for $A(1,2)$ and $B(4,6)$.
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$5$. $d = \sqrt{(4-1)^2+(6-2)^2} = \sqrt{25} = 5$
$5$. $d = \sqrt{(4-1)^2+(6-2)^2} = \sqrt{25} = 5$
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Identify whether $\overline{AB}\cong\overline{CD}$ for $A(0,0),B(3,4),C(1,1),D(4,5)$.
Identify whether $\overline{AB}\cong\overline{CD}$ for $A(0,0),B(3,4),C(1,1),D(4,5)$.
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Yes, both lengths are $5$. Both distances equal $5$, so segments are congruent.
Yes, both lengths are $5$. Both distances equal $5$, so segments are congruent.
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Identify the midpoint of diagonal endpoints $A(0,0)$ and $C(6,2)$.
Identify the midpoint of diagonal endpoints $A(0,0)$ and $C(6,2)$.
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$\left(3,1\right)$. Midpoint of diagonal $AC$.
$\left(3,1\right)$. Midpoint of diagonal $AC$.
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Identify the midpoint of diagonal endpoints $B(2,4)$ and $D(4,-2)$.
Identify the midpoint of diagonal endpoints $B(2,4)$ and $D(4,-2)$.
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$\left(3,1\right)$. Midpoint of diagonal $BD$.
$\left(3,1\right)$. Midpoint of diagonal $BD$.
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Identify whether $A(0,0),B(2,4),C(6,2),D(4,-2)$ form a parallelogram using midpoints.
Identify whether $A(0,0),B(2,4),C(6,2),D(4,-2)$ form a parallelogram using midpoints.
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Yes, diagonals share midpoint $\left(3,1\right)$. Both diagonals have same midpoint.
Yes, diagonals share midpoint $\left(3,1\right)$. Both diagonals have same midpoint.
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Identify the slope of $\overline{AB}$ for $A(0,0)$ and $B(4,0)$.
Identify the slope of $\overline{AB}$ for $A(0,0)$ and $B(4,0)$.
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$m_{AB}=0$. Horizontal segment has slope $0$.
$m_{AB}=0$. Horizontal segment has slope $0$.
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Identify the slope of $\overline{BC}$ for $B(4,0)$ and $C(4,3)$.
Identify the slope of $\overline{BC}$ for $B(4,0)$ and $C(4,3)$.
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Undefined. Vertical segment has undefined slope.
Undefined. Vertical segment has undefined slope.
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Identify whether $\overline{AB}$ is perpendicular to $\overline{BC}$ for $A(0,0),B(4,0),C(4,3)$.
Identify whether $\overline{AB}$ is perpendicular to $\overline{BC}$ for $A(0,0),B(4,0),C(4,3)$.
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Yes, horizontal is perpendicular to vertical. Horizontal and vertical segments are perpendicular.
Yes, horizontal is perpendicular to vertical. Horizontal and vertical segments are perpendicular.
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Identify whether $A(0,0),B(4,0),C(4,3),D(0,3)$ form a rectangle.
Identify whether $A(0,0),B(4,0),C(4,3),D(0,3)$ form a rectangle.
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Yes, opposite sides parallel and adjacent sides perpendicular. Rectangle has opposite sides parallel and right angles.
Yes, opposite sides parallel and adjacent sides perpendicular. Rectangle has opposite sides parallel and right angles.
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Identify whether $A(0,0),B(2,1),C(4,0),D(2,-1)$ form a rectangle using slopes.
Identify whether $A(0,0),B(2,1),C(4,0),D(2,-1)$ form a rectangle using slopes.
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No, adjacent slopes are $\frac{1}{2}$ and $-\frac{1}{2}$ (not perpendicular). Adjacent slopes not negative reciprocals.
No, adjacent slopes are $\frac{1}{2}$ and $-\frac{1}{2}$ (not perpendicular). Adjacent slopes not negative reciprocals.
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Identify whether $A(0,0),B(1,2),C(3,3),D(2,1)$ form a parallelogram using slopes.
Identify whether $A(0,0),B(1,2),C(3,3),D(2,1)$ form a parallelogram using slopes.
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Yes, $m_{AB}=2=m_{CD}$ and $m_{BC}=\frac{1}{2}=m_{AD}$. Opposite sides have equal slopes.
Yes, $m_{AB}=2=m_{CD}$ and $m_{BC}=\frac{1}{2}=m_{AD}$. Opposite sides have equal slopes.
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Identify whether the parallelogram $A(0,0),B(1,2),C(3,3),D(2,1)$ is a rectangle.
Identify whether the parallelogram $A(0,0),B(1,2),C(3,3),D(2,1)$ is a rectangle.
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No, since $m_{AB}m_{BC}=1\neq-1$. Adjacent slopes multiply to $1$, not $-1$.
No, since $m_{AB}m_{BC}=1\neq-1$. Adjacent slopes multiply to $1$, not $-1$.
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Identify $AC$ for $A(0,0)$ and $C(4,0)$.
Identify $AC$ for $A(0,0)$ and $C(4,0)$.
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$4$. Distance between points $A$ and $C$.
$4$. Distance between points $A$ and $C$.
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Identify $BD$ for $B(0,3)$ and $D(4,3)$.
Identify $BD$ for $B(0,3)$ and $D(4,3)$.
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$4$. Distance between points $B$ and $D$.
$4$. Distance between points $B$ and $D$.
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