﻿ Cartesian Coordinates Worksheet | Problems & Solutions Cartesian Coordinates Worksheet

Cartesian Coordinates Worksheet
• Page 1
1.
If (0, 0) (2, 4) (8, 0) ($x$, $y$) are the vertices of a parallelogram, then find the length of the longest diagonal if ($x$ and $y$) are positive integers.  a. 6 units b. 10.77 units c. 14.42 units d. 15.62 units

Solution:

The sides AB and CD are parallel, so the y-coordinate of (2, 4) and (x, y) is same.
[The sides of the parallelogram are parallel.]

y = 4
[y-coordinate of B is 4.]

x = 8 + 2 = 10
[x-coordinate of B is 2.] From the figure and the given coordinates, the vertex (x, y) is (10, 4).

The longest diagonal is AC. The distance between A and C is (x2-x1)²+(y2-y1)²
[Use distance formula.]

= (10-0)²+(4-0)²
[Substitute the values.]

= 100 + 16
[Evaluate the exponents.]

= 116 units
[Simplify.]

The length of the longest diagonal is 116 units.

2.
Find the distance between the points (9, - 3) & (- 4, 2). a. 21.928 units b. 13.928 units c. 16.928 units d. 18.928 units

Solution:

The distance between the points P(x1, y1) and Q(x2, y2) in the coordinate plane is d = (x1-x2)2+(y1-y2)2.
[Formula.]

d = (9 - (-4))2+((-3) - 2)2
[Substitute.]

= 169 + 25
[Evaluate the powers.]

= 194
[Simplify.]

= 13.928 units
[Find the square root.]

3.
Find the midpoint of the line segment joining the points (8, 4) and (- 2, - 2). a. (- 3, 1) b. (3, 1) c. (3, - 1) d. (- 3, - 1)

Solution: The midpoint of the line segment with endpoints (8, 4) and (- 2, - 2) is (8 - 22, 4 - 22).
[Use the formula.]

= (62, 22)

= (3, 1)

4.
Find the distance between the points (2, 4) & (- 7, 2). a. 9.219 units b. 21.219 units c. 17.219 units d. 19.219 units

Solution:

The distance between the points P(x1, y1) and Q(x2, y2) in the coordinate plane is d = (x1-x2)2+(y1-y2)2.
[Formula.]

d = (2 - (-7))2+(4 - 2)2
[Substitute.]

= 81 + 4
[Evaluate the powers.]

= 85
[Simplify.]

= 9.219 units
[Find the square root.]

5.
Find the midpoint of the line segment joining the points (6, 4) and (4, 6). a. (5, - 5) b. (- 5, 5) c. (5, 5) d. (- 5, - 5)

Solution:

The midpoint of the line segment with end points (a, b) and (c, d) is (a + c2, b + d2).
[Formula.]

Midpoint of the line segment with end points (6, 4) and (4, 6) = (6 + 42, 4 + 62)

= (102, 102)

= (5, 5)

6.
Find the perimeter of the triangle determined by the points (- 5, - 3), (0, - 1), (4, 4). a. 14.14 units b. 23.19 units c. 14 units d. 12 units

Solution: Given points are A = (- 5, -3), B = (0, - 1), C = (4, 4) as shown.

Distance between points A and B is AB = (- 5 - 0)2+(-3 + 1)2 = 25 + 4 = 29 = 5.385
[Use formula.]

Distance between points B and C is BC = (0 - 4)2+(- 1 - 4)2 = 16 + 25 = 41 = 6.403
[Use formula.]

Distance between points C and A is CA = (4+5)2+(4+3)2 = 81 + 49 = 130 = 11.402
[Use formula.]

Perimeter of the figure formed by the points = AB + BC + CA

= 5.385 + 6.403 + 11.402

= 23.19

7.
Find the center and radius of the circle.
($x$ - 2)2 + ($y$ - 2)2 = 4 a. (2, - 2); 2 b. (2, 2); 2 c. (2, 2); 4 d. (- 2, 2); 2

Solution:

(x - 2)2 + (y - 2)2 = 4

Center (h, k) = (2, 2)
[Compare with standard form of the circle (x - h)2 + (y - k)2 = r2.]

Radius, r = 4 = 2

8.
Which of the following points is in the third quadrant? a. (- 1, 2) b. (2, - 4) c. (- 4, - 2) d. (9, 9)

Solution:

Any point in the third quadrant should have negative x-coordinate and negative y-coordinate.

Hence, from the given choices, (- 4, - 2) is in the third quadrant.

9.
Evaluate |- 9|. a. - 9 b. ± 9 c. 9 d. None of the above

Solution:

Because -9 < 0, |-9| = - (- 9) = 9
[Use Definition.]

10.
Evaluate |$\pi$ - 12|. a. 8.86 b. ± 8.86 c. 12 d. - 8.86

Solution:

Because π 3.14, π - 12 is negative, so π - 12 < 0.

Thus, |π - 12| = - (π - 12) = 12 - π = 8.86