Partitioning Line Segments via Ratio - Geometry
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Find the point dividing $A(-2,-2)$ to $B(4,10)$ in ratio $1:2$ internally.
Find the point dividing $A(-2,-2)$ to $B(4,10)$ in ratio $1:2$ internally.
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$P(0,2)$. Apply section formula with ratio $1:2$ to both coordinates.
$P(0,2)$. Apply section formula with ratio $1:2$ to both coordinates.
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Which condition on $t$ in $P=A+t(B-A)$ guarantees that $P$ lies between $A$ and $B$?
Which condition on $t$ in $P=A+t(B-A)$ guarantees that $P$ lies between $A$ and $B$?
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$0<t<1$. Point lies between endpoints when parameter is between 0 and 1.
$0<t<1$. Point lies between endpoints when parameter is between 0 and 1.
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Which condition on $t$ makes $P$ beyond $B$ on the ray from $A$ through $B$?
Which condition on $t$ makes $P$ beyond $B$ on the ray from $A$ through $B$?
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$t>1$. Point extends beyond $B$ when parameter exceeds 1.
$t>1$. Point extends beyond $B$ when parameter exceeds 1.
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Which point is closer to $B$ when $P$ divides $\overline{AB}$ internally in ratio $m:n$ with $m>n$?
Which point is closer to $B$ when $P$ divides $\overline{AB}$ internally in ratio $m:n$ with $m>n$?
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$P$ is closer to $B$. Larger first ratio means point is closer to the second endpoint.
$P$ is closer to $B$. Larger first ratio means point is closer to the second endpoint.
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Identify the distance relationship for internal division ratio $m:n$ on $\overline{AB}$ in terms of $AP$ and $PB$.
Identify the distance relationship for internal division ratio $m:n$ on $\overline{AB}$ in terms of $AP$ and $PB$.
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$\frac{AP}{PB}=\frac{m}{n}$. Internal division preserves the ratio of distances.
$\frac{AP}{PB}=\frac{m}{n}$. Internal division preserves the ratio of distances.
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Find $P$ if $A(0,0)$, $B(12,6)$, and $AP:PB=1:3$ internally.
Find $P$ if $A(0,0)$, $B(12,6)$, and $AP:PB=1:3$ internally.
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$P(3,\frac{3}{2})$. Apply section formula with ratio $1:3$ to both coordinates.
$P(3,\frac{3}{2})$. Apply section formula with ratio $1:3$ to both coordinates.
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Find $P$ if $A(2,4)$, $B(14,10)$, and $AP:PB=2:4$ internally (simplify ratio first).
Find $P$ if $A(2,4)$, $B(14,10)$, and $AP:PB=2:4$ internally (simplify ratio first).
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$P(6,6)$. Simplify $2:4$ to $1:2$ then apply section formula.
$P(6,6)$. Simplify $2:4$ to $1:2$ then apply section formula.
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Find $P$ if $A(-3,9)$, $B(9,-3)$, and $AP:PB=1:2$ internally.
Find $P$ if $A(-3,9)$, $B(9,-3)$, and $AP:PB=1:2$ internally.
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$P(1,5)$. Apply section formula with ratio $1:2$ to both coordinates.
$P(1,5)$. Apply section formula with ratio $1:2$ to both coordinates.
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Find $P$ if $A(6,2)$, $B(-2,10)$, and $AP:PB=3:1$ internally.
Find $P$ if $A(6,2)$, $B(-2,10)$, and $AP:PB=3:1$ internally.
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$P(0,8)$. Apply section formula with ratio $3:1$ to both coordinates.
$P(0,8)$. Apply section formula with ratio $3:1$ to both coordinates.
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Find $P$ if directed segment $A(0,0)$ to $B(3,3)$ is partitioned with $t=-1$ in $P=A+t(B-A)$.
Find $P$ if directed segment $A(0,0)$ to $B(3,3)$ is partitioned with $t=-1$ in $P=A+t(B-A)$.
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$P(-3,-3)$. Negative parameter extends point beyond $A$ on the line.
$P(-3,-3)$. Negative parameter extends point beyond $A$ on the line.
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Find $P$ if directed segment $A(1,2)$ to $B(4,8)$ is partitioned with $t=2$ in $P=A+t(B-A)$.
Find $P$ if directed segment $A(1,2)$ to $B(4,8)$ is partitioned with $t=2$ in $P=A+t(B-A)$.
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$P(7,14)$. Parameter greater than 1 extends point beyond $B$.
$P(7,14)$. Parameter greater than 1 extends point beyond $B$.
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Identify the denominator that must be nonzero in the external section formula for ratio $m:n$.
Identify the denominator that must be nonzero in the external section formula for ratio $m:n$.
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$m-n\ne 0$. External division is undefined when $m = n$.
$m-n\ne 0$. External division is undefined when $m = n$.
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State the coordinate form of $P=A+t(B-A)$ when $A(x_1,y_1)$ and $B(x_2,y_2)$ are given.
State the coordinate form of $P=A+t(B-A)$ when $A(x_1,y_1)$ and $B(x_2,y_2)$ are given.
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$P(x_1+t(x_2-x_1),\ y_1+t(y_2-y_1))$. Coordinate form of the vector parametric equation.
$P(x_1+t(x_2-x_1),\ y_1+t(y_2-y_1))$. Coordinate form of the vector parametric equation.
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Find $P$ on $A(0,5)$ to $B(8,1)$ such that $AP:PB=3:1$ internally.
Find $P$ on $A(0,5)$ to $B(8,1)$ such that $AP:PB=3:1$ internally.
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$P(6,2)$. Apply section formula with ratio $3:1$ to both coordinates.
$P(6,2)$. Apply section formula with ratio $3:1$ to both coordinates.
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Find $P$ on $A(-4,-1)$ to $B(2,5)$ such that $AP:PB=1:5$ internally.
Find $P$ on $A(-4,-1)$ to $B(2,5)$ such that $AP:PB=1:5$ internally.
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$P(-3,0)$. Apply section formula with ratio $1:5$ to both coordinates.
$P(-3,0)$. Apply section formula with ratio $1:5$ to both coordinates.
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Find $P$ on $A(7,7)$ to $B(1,-5)$ such that $AP:PB=2:1$ internally.
Find $P$ on $A(7,7)$ to $B(1,-5)$ such that $AP:PB=2:1$ internally.
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$P(3,-1)$. Apply section formula with ratio $2:1$ to both coordinates.
$P(3,-1)$. Apply section formula with ratio $2:1$ to both coordinates.
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Find $P$ on $A(2,-4)$ to $B(8,2)$ such that $AP:PB=1:2$ internally.
Find $P$ on $A(2,-4)$ to $B(8,2)$ such that $AP:PB=1:2$ internally.
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$P(4,-2)$. Apply section formula with ratio $1:2$ to both coordinates.
$P(4,-2)$. Apply section formula with ratio $1:2$ to both coordinates.
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State the vector form for the point $P$ that is fraction $t$ of the way from $A$ to $B$ on a directed segment.
State the vector form for the point $P$ that is fraction $t$ of the way from $A$ to $B$ on a directed segment.
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$P=A+t(B-A)$. Vector form where $t$ is the fractional distance from $A$ to $B$.
$P=A+t(B-A)$. Vector form where $t$ is the fractional distance from $A$ to $B$.
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What is the relationship between ratio $m:n$ (internal) and parameter $t$ in $P=A+t(B-A)$?
What is the relationship between ratio $m:n$ (internal) and parameter $t$ in $P=A+t(B-A)$?
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$t=\frac{m}{m+n}$. Parameter $t$ equals the first part over the total.
$t=\frac{m}{m+n}$. Parameter $t$ equals the first part over the total.
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What is the relationship between parameter $t$ and ratio $m:n$ for internal division from $A$ toward $B$?
What is the relationship between parameter $t$ and ratio $m:n$ for internal division from $A$ toward $B$?
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$m:n=t:(1-t)$. Ratio converts to parameter by using $t$ and its complement.
$m:n=t:(1-t)$. Ratio converts to parameter by using $t$ and its complement.
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Which condition on $t$ makes $P$ on the ray opposite $B$ from $A$ along line $AB$?
Which condition on $t$ makes $P$ on the ray opposite $B$ from $A$ along line $AB$?
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$t<0$. Negative parameter places point on opposite side of $A$.
$t<0$. Negative parameter places point on opposite side of $A$.
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Identify the key idea of the section formula: $P$ is a weighted average of which points?
Identify the key idea of the section formula: $P$ is a weighted average of which points?
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$A(x_1,y_1)$ and $B(x_2,y_2)$. Section formula creates weighted averages of the endpoint coordinates.
$A(x_1,y_1)$ and $B(x_2,y_2)$. Section formula creates weighted averages of the endpoint coordinates.
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Find the point dividing $A(0,0)$ to $B(10,0)$ in ratio $1:4$ internally.
Find the point dividing $A(0,0)$ to $B(10,0)$ in ratio $1:4$ internally.
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$P(2,0)$. Apply section formula: $\frac{1 \cdot 10 + 4 \cdot 0}{1+4} = 2$.
$P(2,0)$. Apply section formula: $\frac{1 \cdot 10 + 4 \cdot 0}{1+4} = 2$.
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Find the point dividing $A(0,0)$ to $B(10,0)$ in ratio $3:2$ internally.
Find the point dividing $A(0,0)$ to $B(10,0)$ in ratio $3:2$ internally.
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$P(6,0)$. Apply section formula: $\frac{3 \cdot 10 + 2 \cdot 0}{3+2} = 6$.
$P(6,0)$. Apply section formula: $\frac{3 \cdot 10 + 2 \cdot 0}{3+2} = 6$.
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Find the point dividing $A(0,0)$ to $B(0,12)$ in ratio $1:2$ internally.
Find the point dividing $A(0,0)$ to $B(0,12)$ in ratio $1:2$ internally.
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$P(0,4)$. Apply section formula: $\frac{1 \cdot 12 + 2 \cdot 0}{1+2} = 4$.
$P(0,4)$. Apply section formula: $\frac{1 \cdot 12 + 2 \cdot 0}{1+2} = 4$.
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Find the point dividing $A(-4,0)$ to $B(6,0)$ in ratio $2:3$ internally.
Find the point dividing $A(-4,0)$ to $B(6,0)$ in ratio $2:3$ internally.
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$P(0,0)$. Apply section formula: $\frac{2 \cdot 6 + 3 \cdot (-4)}{2+3} = 0$.
$P(0,0)$. Apply section formula: $\frac{2 \cdot 6 + 3 \cdot (-4)}{2+3} = 0$.
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Find the point dividing $A(2,2)$ to $B(8,8)$ in ratio $1:2$ internally.
Find the point dividing $A(2,2)$ to $B(8,8)$ in ratio $1:2$ internally.
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$P(4,4)$. Apply section formula with both coordinates using ratio $1:2$.
$P(4,4)$. Apply section formula with both coordinates using ratio $1:2$.
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What is the midpoint formula as a special case of partitioning $A(x_1,y_1)$ to $B(x_2,y_2)$?
What is the midpoint formula as a special case of partitioning $A(x_1,y_1)$ to $B(x_2,y_2)$?
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$M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Special case when $m=n=1$ in the section formula.
$M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Special case when $m=n=1$ in the section formula.
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State the section formula for point $P$ dividing directed segment $A(x_1,y_1)$ to $B(x_2,y_2)$ in ratio $m:n$ (external).
State the section formula for point $P$ dividing directed segment $A(x_1,y_1)$ to $B(x_2,y_2)$ in ratio $m:n$ (external).
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$P\left(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n}\right)$. External division uses subtraction in numerator and denominator.
$P\left(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n}\right)$. External division uses subtraction in numerator and denominator.
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Identify the ratio $m:n$ that produces the midpoint of a segment using the section formula.
Identify the ratio $m:n$ that produces the midpoint of a segment using the section formula.
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$1:1$. Equal division creates the midpoint of any segment.
$1:1$. Equal division creates the midpoint of any segment.
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