Suppose you have a heat segment PQ ¯ on the name: coordinates plane, and you require to find the allude on the segment 1 3 that the method from ns to Q.

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

The length of the line is 6 units and also the suggest on the segment 1 3 of the way from p to Q would be 2 systems away from P, 4 devices away from Q and would be at ( 2,0 ).

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

The components of the command segment PQ ¯ are 〈 6,3 〉 and also we require to uncover the point, speak X on the segment 1 3 of the way from p to Q.

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

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

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

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

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

now in a similar way, the components of the segment PX ¯ where X is a point on the segment 1 3 of the way from ns to Qare 〈 ( 1 3 )( 6 ),( 1 3 )( −4 ) 〉=〈 2,−1.25 〉.

To discover the works with of the point X add the materials of the segment PX ¯ to the collaborates of the initial suggest P.

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

Note the the result segments, PX ¯ and XQ ¯ , have lengths in a ratio of 1:2.

In general: what if you need to uncover a suggest on a line segment the divides it right into two segments v lengths in a proportion a:b?

think about the directed line segment XY ¯ with coordinates of the endpoints together X( x 1 , y 1 ) and Y( x 2 , y 2 ).

Suppose the suggest Z separated the segment in the ratio a:b, then the point is a a+b that the way from X come Y.

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

Then, the X-coordinate that the point 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 suggest 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 suggest that divides the directed line segment MN ¯ with the coordinates of endpoints in ~ M( −4,0 ) and also M( 0,4 ) in the proportion 3:1?

allow L be the point that divides MN ¯ in the proportion 3:1.

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

instead of in the formula. The works with 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 allude L( −1,3 ) divides MN ¯ in the proportion 3:1.

**Example 2:**

What room the collaborates of the allude that divides the directed heat segment ab ¯ in the ratio 2:3?

Let C it is in the suggest that divides abdominal ¯ in the ratio 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 collaborates of C are

( 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 )