Question

A treasure map says that a treasure is buried so that it partitions the distance between a rock and a tree in a 5:9 ratio. Marina traced the map onto a coordinate plane to find the exact location of the treasure.
What are the coordinates of the treasure? If necessary, round the coordinates to the nearest tenth.

(11.4, 14.2)
(7.6, 8.8)
(5.7, 7.5)
(10.2, 12.6)

Answer 1

Answer:

The coordinates of Treasure round to the nearest tenth is:

(7.6,8.8)

Step-by-step explanation:

We are given the coordinates of rock as: (3,2)

and Tree as: (16,21)

Now as Treasure divides the line segment between rock and tree in the ratio 5:9.

As we know if any point C(e,f) divides the line segment A(a,b) and B(c,d) in the ratio m:n then the coordinates of point C is calculated as:

$e=\dfrac{m\times c+n\times a}{m+n}$

and $f=\dfrac{m\times d+n\times b}{m+n}$

Here we have (a,b)=(3,2)

(c,d)=(16,21)

and m:n=5:9.

Let the coordinates of Treasure are (e,f).

$e=\dfrac{5\times 16+9\times 3}{5+9}\\\\e=\dfrac{80+27}{14}\\\\e=\dfrac{107}{14}\\\\e=7.6$

and $f=\dfrac{5\times 21+9\times 2}{5+9}\\\\f=\dfrac{105+18}{14}\\\\f=\dfrac{123}{14}\\\\f=8.8$

Hence, the coordinates of Treasure are:

(7.6,8.8).

Answer 2

Let the coordinates of the treasure be (x,y).
Let $d_{1}$ =  distance from the rock to the treasure.
Let $d_{2}$ =  distance from the tree to the treasure.
 $d_{1} = \sqrt{(x\2 + (y-2)^2}$
$d_{2} = \sqrt{(16-x)^2 + (21-y)^2}$
We want $\frac{ d_{1} }{ d_{2} }$ = 5/9 in order to locate the treasure. 

Note that 5/9 = 0.556 (approx)
Test (11.4, 14.2)
d1 = 14.8, d2 = 8.2, d1/d2 = 1.8           Incorrect
Test (7.6, 8.8)
d1 = 8.2,  d2 = 14.8,  d1/d2 = 0.554    Correct
Test (5.7, 7.5)
d1 = 6.13,  d2 = 16.98,  d1/d2 = 0.36   Incorrect
Test (10.2, 12.6)
d1 = 12.8,  d2 = 10.2,  d1/d2 = 1.255   Incorrect

Answer: The correct answer is (7.6, 8.8)