Here is a scale drawing of a garden. Kim wants to plant a tree in the garden according to the following rules:
It must be 7 m from A and 4 m from C.
Place a cross where Kim can plant the tree.
1 cm represents 2 m
D
2.5 cm
А A
B.
4 cm
View One Minute Version

The coordinates of the cross on the scale drawing which locates where Kim can plant the tree is (3.45 cm, 0.58 cm)

Step-by-step explanation:

The question relates to coordinates geometry and the solution can be obtained graphically by construction as shown in the attached drawing

The given parameters of the scale drawing are;

The scale of the drawing is 1 cm = 2 m

The dimensions of the rectangle representing the garden in the scale drawing is given as follows;

The width of the rectangle = 2.5 cm

The length of the rectangle = 4 cm

The required location where the tree is to be planted = 7 m from A and 4 m from point C

Therefore, the location where Kim can plant the tree on the scale drawing is 3.5 cm from point A and 2 cm from point C

Taking the coordinate of point A as the origin, we have;

The coordinates of point A = (0, 0)

∴ The coordinates of point C = (4, 2.5)

The general equation of a circle is (x - h)² + (y - k)² = r²

Where, (h, k) is the coordinate of the center of the circle

The location of the point is on a radius 3.5 cm from point A and a radius 2 cm from point C

Therefore, for point A, we have;

(x - 0)² + (y - 0)² = 3.5²

x² + y² = 12.25...(1)

For point C, we have;

(x - 4)² + (y - 2.5)² = 2² = 4

Which gives;

x² - 8·x + 16 + y² - 5·y + 6.25 = 4

x² + y² - 8·x - 5·y + 22.25 = 4

We have x² + y² = 12.25 from equation (1), therefore, we have;

12.25 - 8·x - 5·y + 22.25 = 4

- 8·x - 5·y = 4 - 22.25 - 12.25 = -30.5

∴ 8·x + 5·y = 30.5

y = (30.5 - 8·x)/5

Therefore, we have;

x² + y² = 12.25 = x² + ((30.5 - 8·x)/5)² = 12.25

Using a graphing calculator we get;

356·x² - 1952·x + 2496 = 0

Which gives;

x = (1952 ± √((-1952)² - 4×356×2496))/(2 × 356)

x = 3.452197227 or 2.03094884081

To two decimal places, we have the coordinates of the points where the tree can be planted given as follows;

Therefore, y = (30.5 - 8×3.452197227)/5 = 0.5764844368

or y = (30.5 - 8×2.03094884081)/5 = 2.8504818547

y = 0.5764844368 or 2.8504818547

To two decimal places, we have the coordinates of the places where Kim can plant the tree are given as follows;

(3.45, 0.58) or (2.03, 2.85)

We note that the point (2.03, 2.85) is at the outside the drawing the area as 2.85 cm is more than 2.5 cm which is the width (height) of the rectangle as shown in the attached drawing created with Microsoft Word.

Therefore, the coordinates of the cross where Kim can plant the tree is (3.45, 0.58)