x & = -3 + \frac{1}{3} \times \big(3 - (-3)\big) \\ If point P(x,y)P (x,y)P(x,y) lies on line segment AB‾\overline{AB}AB (((between points AAA and B)B)B) and satisfies AP:PB=m:n,AP:PB=m:n,AP:PB=m:n, then we say that PPP divides AB‾\overline{AB}AB internally in the ratio m:n.m:n.m:n. The point of division has the coordinates. A reflection is defined using a line. When measured parallel to the xxx-axis, we get, x=−3+13×(3−(−3))=−1.\begin{aligned} Directed line segments C C prime, D D prime, E E prime and v are all the same length and direction, slanting slightly downward and to the right, with the endpoint on the top end and the arrow at the bottom end. The section formula builds on it and is a more powerful tool; it locates the point dividing the line segment in any desired ratio. Given A=(−2,−1)A= (-2,-1)A=(−2,−1) and B=(4,11)B=(4,11)B=(4,11), point P=(x,y)P= (x,y)P=(x,y) internally divides line segment AB‾\overline{AB}AB in the ratio m:nm:nm:n. If PP P is the intersection point of AB‾\overline{AB}AB and the yyy-axis, what is the value of m:n?m : n?m:n? □P (x,y) = \left( \dfrac { m{ x }_{ 2 }+n{ x }_{ 1 } }{ m+n }, \dfrac { m{ y }_{ 2 }+n{ y }_{ 1 } }{ m+n } \right).\ _\squareP(x,y)=(m+nmx2​+nx1​​,m+nmy2​+ny1​​). In the figure, $$A'$$ is the image of $$A$$ under the reflection across the line $$m$$. To find the point that’s two-thirds of the distance from (–4,1) to the other endpoint, (8,7): Replace x1 with –4, x2 with 8, y1 with 1, y2 with 7, and k with 2/3. (4), P(x,y)=(mx2−nx1m−n,my2−ny1m−n). & = \frac{(m - n)x_1 + mx_2 - mx_1 }{m-n} \\ P(3,0) 2. Simplify. The arrow of the directed line segment specifies the direction of the translation, and the length of the directed line segment specifies how far the figure gets translated. Derive a formula that calculates the midpoint of the segment connecting (x 1, y 1) with (x 2, y 2). A line segment with an arrow at one end specifying a direction. 1 (i) and (ii) are both directed lines. \qquad (4)y=m−nmy2​−ny1​​. Let us find the lengths of aaa and b:b:b: a=(−3)−(−5)=2,b=4−(−3)=7.a = (-3) - (-5) = 2, \quad b = 4 - (-3) = 7.a=(−3)−(−5)=2,b=4−(−3)=7. The thing you should remember is that PPP divides ABABAB in the ratio 2:12 : 12:1 and QQQ divides ABABAB in the ratio 1:21 : 21:2. (3)\begin{aligned} Since point PPP is on the yyy-axis, its xxx coordinate is zero. To relate this to a dilation it means that we will be doing a reduction (0 < k < 1) so that the point will be on the segment. VERY IMPORTANT -- In the first example you might notice that WE DID NOT REDUCE THE SLOPE!!! P=(mx2−nx1m−n,my2−ny1m−n).P=\left( \dfrac{mx_2 - nx_1}{m-n}, \dfrac{my_2 - ny_1}{m-n} \right) .P=(m−nmx2​−nx1​​,m−nmy2​−ny1​​). Magnitude refers to the length of the directed line segment and is usually based on a scale.

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