Suppose you have actually a heat segment PQ ¯ on the coordinate plane, and also you require to find the point on the segment 1 3 of the way from ns to Q.

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Let’s first take the easy instance where p is in ~ the origin and also line segment is a horizontal one.

The length of the heat is 6 units and the allude on the segment 1 3 the the means from ns to Q would be 2 systems away native P, 4 units away from Q and also would be at ( 2,0 ).

Consider the case where the segment is not a horizontal or upright line.

The components of the directed segment PQ ¯ space 〈 6,3 〉 and also we require to discover the point, to speak X ~ above the segment 1 3 the the way from ns to Q.

Then, the materials of the segment PX ¯ space 〈 ( 1 3 )( 6 ),( 1 3 )( 3 ) 〉=〈 2,1 〉.

because the initial allude of the segment is at origin, the collaborates of the suggest X are given by ( 0+2,0+1 )=( 2,1 ).

now let’s do a trickier problem, wherein neither p nor Q is at the origin.

Use the end points that the segment PQ ¯ to write the materials of the directed segment.

〈 ( x 2 − x 1 ),( y 2 − y 1 ) 〉=〈 ( 7−1 ),( 2−6 ) 〉                                             =〈 6,−4 〉

currently in a comparable way, the components of the segment PX ¯ whereby X is a suggest on the segment 1 3 that the method from ns to Qare 〈 ( 1 3 )( 6 ),( 1 3 )( −4 ) 〉=〈 2,−1.25 〉.

To discover the collaborates of the suggest X add the components of the segment PX ¯ come the works with of the initial suggest P.

So, the coordinates of the allude X space ( 1+2,6−1.25 )=( 3,4.75 ).

Note that the resulting segments, PX ¯ and also XQ ¯ , have lengths in a proportion of 1:2.

In general: what if you need to find a allude on a line segment the divides it into two segments with lengths in a ratio a:b?

take into consideration the directed heat segment XY ¯ with works with of the endpoints as X( x 1 , y 1 ) and also Y( x 2 , y 2 ).

Suppose the suggest Z split the segment in the ratio a:b, then the point is a a+b of the way from X to Y.

So, generalizing the an approach we have, the components of the segment XZ ¯ are 〈 ( a a+b ( x 2 − x 1 ) ),( a a+b ( y 2 − y 1 ) ) 〉.

Then, the X-coordinate the the suggest Z is

x 1 + a a+b ( x 2 − x 1 )= x 1 ( a+b )+a( x 2 − x 1 ) a+b                                         = b x 1 +a x 2 a+b .

Similarly, the Y-coordinate is

y 1 + a a+b ( y 2 − y 1 )= y 1 ( a+b )+a( y 2 − y 1 ) a+b                                         = b y 1 +a y 2 a+b .

Therefore, the works with of the point Z space ( b x 1 +a x 2 a+b , b y 1 +a y 2 a+b ).

Example 1:

Find the coordinates of the allude that divides the directed line segment MN ¯ with the collaborates of endpoints at M( −4,0 ) and also M( 0,4 ) in the proportion 3:1?

permit L it is in the suggest that divides MN ¯ in the proportion 3:1.

Here, ( x 1 , y 1 )=( −4,0 ),( x 2 , y 2 )=( 0,4 ) and also a:b=3:1.

instead of in the formula. The collaborates of L space

( 1( −4 )+3( 0 ) 3+1 , 1( 0 )+3( 4 ) 3+1 ).

Simplify.

( −4+0 4 , 0+12 4 )=( −1,3 )

Therefore, the suggest L( −1,3 ) divides MN ¯ in the ratio 3:1.

Example 2:

What are the coordinates of the suggest that divides the directed heat segment abdominal ¯ in the ratio 2:3?

Let C it is in the allude that divides abdominal ¯ in the proportion 2:3.

Here, ( x 1 , y 1 )=( −4,4 ),( x 2 , y 2 )=( 6,−5 ) and also a:b=2:3.

Substitute in the formula. The coordinates of C space

( 3( −4 )+2( 6 ) 5 , 3( 4 )+2( −5 ) 5 ).

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Simplify.

( −12+12 5 , 12−10 5 )=( 0, 2 5 )                                             =( 0,0.4 )