Question

Geometry

Answer 1

Answer: (5.2, 8)

Step-by-step explanation:

Given : G(1,2) and K (8,12).

To find :  The coordinates of P that partitions gk in the ratio of 3:2

Section formula :

The line segment having endpoints (a,b) and (c,d) is divided in ration m:n by point M , then the coordinates of the M will be :-

$x=\dfrac{mc+na}{m+n}\ ;\ y=\dfrac{md+nb}{m+n}$

Similarly,

$x=\dfrac{3(8)+2(1)}{3+2}\ ;\ y=\dfrac{3(12)+2(2)}{3+2}$

Now simplify , we get

x=5.2  and y=8

Hence, the coordinates of P that partitions GK in the ratio of 3:2 = (5.2, 8)

Answer 2

A point P on the line segment GK may be parametrically represented as

$P = (1-t)G + tK$

t is a real parameter which controls where on the segment P is.  t=0 means P=G, t=1 means P=K.  We're interested in the t that gives a 3:2 ratio for PG:PK.  That's closer to K so t>1/2.   t=3/(3+2) = 3/5.

$P=(1-3/5)(1,2) + (3/5)(8,12) = (2/5,4/5)+(24/5,36/5)=(26/5,40/5)=(26/5,8)$

Answer: (26/5, 8)