In a coordinate plane, what is the slope of the line passing through points B(5, 8) and D(6, 1)?

Correct Option: A. Solution: The slope is calculated as $$\frac{1-8}{6-5} = -7$$.

Coordinate Geometry

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Summary

Chapter Summary

Key Points

Distance Formula: The distance between two points P(x₁, y₁) and Q(x₂, y₂) is given by:
- $d = \sqrt{(x₂ - x₁)² + (y₂ - y₁)²}$
Distance from Origin: The distance of a point P(x, y) from the origin is:
- $d = \sqrt{x² + y²}$
Section Formula: The coordinates of the point P(x, y) that divides the line segment joining A(x₁, y₁) and B(x₂, y₂) in the ratio m₁:m₂ are:
- $x = \frac{m₁x₂ + m₂x₁}{m₁ + m₂}, \quad y = \frac{m₁y₂ + m₂y₁}{m₁ + m₂}$
Mid-point Formula: The mid-point of the line segment joining A(x₁, y₁) and B(x₂, y₂) is:
- $P = \left( \frac{x₁ + x₂}{2}, \frac{y₁ + y₂}{2} \right)$

Examples

Example of Distance Calculation: To find the distance between points (2, 3) and (4, 1), apply the distance formula.
Example of Section Formula: To find the coordinates of the point dividing the segment joining (-1, 7) and (4, -3) in the ratio 2:3, use the section formula.

Applications

Coordinate geometry is widely applied in fields such as physics, engineering, navigation, and art.

Learning Objectives

Understand the concept of coordinate geometry.
Apply the distance formula to find the distance between two points.
Use the section formula to find coordinates of points dividing line segments.
Identify collinear points using distance calculations.
Determine the type of quadrilateral formed by given points.
Solve problems involving distances and ratios in coordinate geometry.

Detailed Notes

Chapter 7: Coordinate Geometry

7.1 Introduction

To locate a point on a plane, a pair of coordinate axes is required.
The x-coordinate (abscissa) is the distance from the y-axis.
The y-coordinate (ordinate) is the distance from the x-axis.
Points on the x-axis are of the form (x, 0) and on the y-axis are of the form (0, y).

7.2 Distance Formula

The distance between two points P(x₁, y₁) and Q(x₂, y₂) is given by:

$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
The distance of a point P(x, y) from the origin O(0, 0) is:

$OP = \sqrt{x^2 + y^2}$

Examples

Example 1: Find the distance between points (2, 3) and (4, 1).
Example 2: Find the distance between points (0, 0) and (36, 15).

7.3 Section Formula

The coordinates of the point P(x, y) that divides the line segment joining points A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ are:

$x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \quad y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}$

Examples

Example 6: Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1.
- Solution: (7, 3)
Example 7: Find the ratio in which the point (-4, 6) divides the line segment joining A(-6, 10) and B(3, -8).
- Solution: m₁ : m₂ = 2 : 7

Exercises

Find the distance between the following pairs of points:
- (2, 3), (4, 1)
- (-5, 7), (-1, 3)
Find the coordinates of the point which divides the join of (-1, 7) and (4, -3) in the ratio 2:3.
Determine if the points (1, 5), (2, 3), and (-2, -11) are collinear.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Misapplication of Distance Formula: Students often forget to square the differences in coordinates when using the distance formula, leading to incorrect calculations.
Incorrect Use of Section Formula: Failing to correctly apply the section formula can result in wrong coordinates for points dividing line segments.
Assuming Collinearity: Students may incorrectly assume points are collinear without verifying using the distance formula or slope calculations.
Ignoring Units: When calculating distances, students sometimes neglect to include units, which can lead to confusion in answers.

Tips for Success

Double-Check Calculations: Always recheck your calculations for distance and coordinates, especially when applying formulas.
Visualize Problems: Draw diagrams to help visualize the problem, especially for coordinate geometry questions.
Practice with Examples: Work through multiple examples of distance and section formula problems to build confidence.
Understand Concepts: Make sure to understand the underlying concepts of distance and section formulas rather than just memorizing them.

Practice & Assessment

Multiple Choice Questions

Yes, it is a square.

No, it is not a square.

Yes, it is a rectangle.

No, it is a parallelogram.

Correct Answer: B

Solution:

To determine if ABCD is a square, calculate the distances AB, BC, CD, and DA. If all sides are equal and the diagonals are equal, it is a square. Calculations show that the sides are not equal, so it is not a square.

4 meters

5 meters

6 meters

7 meters

Correct Answer: A

Solution:

The distance between the green flag at the 4th line and the red flag at the 8th line is

8 - 4 = 4

meters.

5 units

6 units

7 units

8 units

Correct Answer: A

Solution:

Using the distance formula, the distance between points A and B is

\sqrt{(5-2)^2 + (8-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

units.

(5, 7)

(4, 6)

(5, 6)

(6, 7)

Correct Answer: A

Solution:

The midpoint

M

of a line segment with endpoints

(x_1, y_1)

and

(x_2, y_2)

is given by

M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

. For

P(3, 5)

and

Q(7, 9)

M = \left( \frac{3 + 7}{2}, \frac{5 + 9}{2} \right) = (5, 7)

0.3 M

0.4 M

0.1 M

0.2 M

Correct Answer: B

Solution:

The stoichiometry of the reaction is

2A + B \rightarrow C + D

. At equilibrium,

[C] = 0.2

M, which means

0.2

M of

A

and

0.1

M of

B

have reacted (since

2A

is required for every

B

). Therefore, the equilibrium concentration of

A

0.5 - 0.2 = 0.3

39 km

41 km

45 km

51 km

Correct Answer: B

Solution:

Using the Pythagorean theorem, the hypotenuse is calculated as:

\sqrt{36^2 + 15^2} = \sqrt{1296 + 225} = \sqrt{1521} = 39

km.

Correct Answer: B

Solution:

The distance between two points

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. For points A(2, 4) and C(9, 4), the distance is

\sqrt{(9 - 2)^2 + (4 - 4)^2} = \sqrt{7^2} = 7

15 km

36 km

51 km

39 km

Correct Answer: B

Solution:

Point B is directly 36 km east of point A, so the distance is 36 km.

3 units

5 units

6 units

7 units

Correct Answer: D

Solution:

The distance between points A and B is calculated using the distance formula:

\sqrt{(5 - 2)^2 + (8 - 4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

units.

-7

-1

Correct Answer: A

Solution:

The slope is calculated as

\frac{1-8}{6-5} = -7

Rectangle

Square

Rhombus

Trapezoid

Correct Answer: A

Solution:

To determine the type of quadrilateral, calculate the distances between the points: AB = 5, BC = 5, CD = 5, DA = 5, and diagonals AC = 7, BD = 7. Since opposite sides are equal and diagonals are equal, the quadrilateral is a rectangle.

5 units

7 units

9 units

11 units

Correct Answer: B

Solution:

The distance between points A and C is calculated using the distance formula:

\sqrt{(9 - 2)^2 + (4 - 4)^2} = \sqrt{7^2} = 7

units.

Square

Rectangle

Rhombus

Trapezoid

Correct Answer: D

Solution:

The points form a trapezoid as only one pair of opposite sides are parallel.

15 units

13 units

10 units

18 units

Correct Answer: A

Solution:

According to the Pythagorean theorem, the hypotenuse

c

is given by

c = \sqrt{a^2 + b^2}

, where

a

and

b

are the lengths of the legs. Here,

c = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15

units.

-1

0.75

-0.75

Correct Answer: D

Solution:

The slope of the line passing through points A and D is

\frac{1-4}{6-2} = \frac{-3}{4} = -0.75

1 mole

2 moles

3 moles

4 moles

Correct Answer: B

Solution:

The balanced chemical equation for the reaction is

2H_2 + O_2 \rightarrow 2H_2O

. According to the equation, 2 moles of hydrogen gas react with 1 mole of oxygen gas to produce 2 moles of water.

(5.5, 4)

(6, 4)

(5, 4)

(4.5, 4)

Correct Answer: A

Solution:

The midpoint of a line segment joining

(x_1, y_1)

and

(x_2, y_2)

is given by

\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

. Substituting the values, we get

\left(\frac{2 + 9}{2}, \frac{4 + 4}{2}\right) = (5.5, 4)

7 units

5 units

9 units

6 units

Correct Answer: A

Solution:

The horizontal distance between two points with the same y-coordinate is the difference in their x-coordinates:

9 - 2 = 7

units.

(4.5, 2.5)

(4, 2)

(5, 3)

(3, 3)

Correct Answer: A

Solution:

The midpoint of a line segment with endpoints

(x_1, y_1)

and

(x_2, y_2)

is given by

\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

. For points P(3, 1) and Q(6, 4), the midpoint is

\left(\frac{3 + 6}{2}, \frac{1 + 4}{2}\right) = (4.5, 2.5)

Rectangle

Square

Rhombus

Trapezoid

Correct Answer: D

Solution:

To determine the type of quadrilateral, calculate the distances: AB = 5, BC = 5, CD = 5, DA = 5, AC = 7, BD = 7. The opposite sides are not equal, hence it's not a rectangle or square. It forms a trapezoid as only one pair of opposite sides are parallel.

5 units

6 units

7 units

8 units

Correct Answer: A

Solution:

Using the distance formula:

\sqrt{(6-2)^2 + (1-4)^2} = \sqrt{16 + 9} = 5

units.

Correct Answer: C

Solution:

In a full binary tree, the number of total nodes

N

is given by

N = 2L - 1

, where

L

is the number of leaf nodes. Here,

L = 15

, so

N = 2 \times 15 - 1 = 30 - 1 = 31

Correct Answer: A

Solution:

The distance between two points (x1, y1) and (x2, y2) is given by the formula:

\sqrt{(x2 - x1)^2 + (y2 - y1)^2}

. Substituting the coordinates, we get:

\sqrt{(6 - 0)^2 + (8 - 0)^2} = \sqrt{36 + 64} = \sqrt{100} = 10

1 kcal

10 kcal

100 kcal

1000 kcal

Correct Answer: B

Solution:

Energy transfer efficiency is 10% at each trophic level. Primary producers have 1000 kcal, primary consumers receive 10% of this, which is 100 kcal. Secondary consumers receive 10% of 100 kcal, which is 10 kcal. Tertiary consumers receive 10% of 10 kcal, which is 1 kcal.

3 units

5 units

6 units

7 units

Correct Answer: D

Solution:

Using the distance formula, the distance between points C and D is

\sqrt{(9-6)^2 + (4-1)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}

units, which is approximately 7 units.

Increased growth rate during the day

Reduced competition for sunlight

Enhanced water conservation

Improved resistance to herbivores

Correct Answer: B

Solution:

Photosynthesizing at night would reduce competition for sunlight, as the plant can utilize light energy at a time when other plants are not. This allows the plant to occupy ecological niches where light is limited during the day.

\frac{4}{3}

\frac{3}{4}

1

2

Correct Answer: A

Solution:

The slope of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

is given by

\frac{y_2 - y_1}{x_2 - x_1}

. Substituting the values, we get

\frac{8 - 4}{5 - 2} = \frac{4}{3}

Champa is correct; ABCD is a square.

Chameli is correct; ABCD is not a square.

Both are incorrect; ABCD is a rectangle.

Both are correct; ABCD is a rhombus.

Correct Answer: B

Solution:

The distances between the points do not satisfy the conditions for a square, as the sides and diagonals are not equal.

Correct Answer: B

Solution:

The vertical distance between points B and D is the absolute difference in their y-coordinates:

|8 - 1| = 7

Rectangle

Rhombus

Parallelogram

Trapezoid

Correct Answer: D

Solution:

The quadrilateral is a trapezoid because only one pair of opposite sides (AB and CD) is parallel.

Correct Answer: C

Solution:

The horizontal distance between points A and C is the difference in their x-coordinates:

9 - 2 = 7

Rectangle

Square

Rhombus

Trapezoid

Correct Answer: A

Solution:

To determine the type of quadrilateral, calculate the distances:

PQ = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{25} = 5

QR = \sqrt{(7-4)^2 + (2-6)^2} = \sqrt{25} = 5

RS = \sqrt{(7-4)^2 + (2+2)^2} = \sqrt{25} = 5

, and

SP = \sqrt{(4-1)^2 + (-2-2)^2} = \sqrt{25} = 5

. All sides are equal, and diagonals

PR = \sqrt{(7-1)^2 + (2-2)^2} = 6

and

QS = \sqrt{(4-4)^2 + (6+2)^2} = 8

. Since diagonals are not equal, the quadrilateral is a rectangle.

Square

Rectangle

Rhombus

Trapezoid

Correct Answer: D

Solution:

The points do not form a square or rectangle as the sides are not equal and parallel. They form a trapezoid because only one pair of sides is parallel.

7 units

5 units

9 units

11 units

Correct Answer: A

Solution:

The length of AC is calculated as the difference in x-coordinates:

9 - 2 = 7

units.

2 moles

3 moles

4 moles

1 mole

Correct Answer: A

Solution:

The reaction is

2A + B \rightarrow C

. Starting with 4 moles of

A

and 2 moles of

B

, the limiting reagent is

A

because

4

moles of

A

can react with

2

moles of

B

to produce

2

moles of

C

. Therefore, 2 moles of

C

can be produced.

Square

Rectangle

Rhombus

Trapezium

Correct Answer: D

Solution:

Using the distance formula, AB = 5, BC = 5, CD = 5, and DA = 5. However, the diagonals AC and BD are not equal, so ABCD is not a square or rectangle. It is a trapezium.

9 square units

12 square units

15 square units

18 square units

Correct Answer: A

Solution:

Using the formula for the area of a triangle with vertices at (x1, y1), (x2, y2), (x3, y3):

\frac{1}{2} \left| x1(y2-y3) + x2(y3-y1) + x3(y1-y2) \right|

, the area is

\frac{1}{2} \left| 3(4-6) + 6(6-1) + 9(1-4) \right| = \frac{1}{2} \left| -6 + 30 - 27 \right| = \frac{1}{2} \times 9 = 9

square units.

C(9, 4)

D(6, 1)

E(5, 4)

F(3, 8)

Correct Answer: A

Solution:

Point C(9, 4) forms a right triangle with points A(2, 4) and B(5, 8) as the horizontal and vertical distances create perpendicular sides.

7 units

8 units

9 units

10 units

Correct Answer: C

Solution:

The distance between points B and D is calculated using the distance formula:

\sqrt{(6 - 5)^2 + (1 - 8)^2} = \sqrt{1 + 49} = \sqrt{50} = 7.07

units, approximately 9 units.

3 meters from the starting point

5 meters from the starting point

7 meters from the starting point

10 meters from the starting point

Correct Answer: B

Solution:

The midpoint between the green flag at 2 meters and the red flag at 8 meters is given by

(2 + 8) / 2 = 5

meters.

36 km

15 km

39 km

51 km

Correct Answer: C

Solution:

The distance between two points

(x_1, y_1)

and

(x_2, y_2)

is given by

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. Here, it is

\sqrt{(36 - 0)^2 + (15 - 0)^2} = \sqrt{1296 + 225} = \sqrt{1521} = 39

km.

50 ms

5 ms

500 ms

100 ms

Correct Answer: A

Solution:

The time constant

\tau

of an RC circuit is given by

\tau = RC

. Substituting the given values,

\tau = 10 \Omega \times 5 \times 10^{-6} F = 50 \times 10^{-3} s = 50 ms

5 units

6 units

7 units

8 units

Correct Answer: A

Solution:

The distance is calculated using the distance formula:

\sqrt{(7 - 3)^2 + (5 - 1)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.66

, rounded to 5 units.

5

4

3

6

Correct Answer: A

Solution:

The distance between two points

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. Substituting the values, we get

\sqrt{(5 - 2)^2 + (8 - 4)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

3 units

5 units

8 units

4 units

Correct Answer: A

Solution:

Since point O is at (0,0) and point C is at (0,3), the length of OC is the difference in the y-coordinates, which is 3 units.

5

3\sqrt{2}

\sqrt{18}

\sqrt{34}

Correct Answer: D

Solution:

Using the distance formula, the length of line segment CD is

\sqrt{(6 - 9)^2 + (1 - 4)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18}

\sqrt{52}

\sqrt{68}

\sqrt{100}

\sqrt{34}

Correct Answer: B

Solution:

The length of diagonal AC is calculated using the distance formula:

\sqrt{(9-2)^2 + (4-4)^2} = \sqrt{7^2 + 0^2} = \sqrt{49} = 7

. However, this was incorrect in the options. Correct calculation should be

\sqrt{(9-2)^2 + (4-4)^2} = \sqrt{49} = 7

Square

Rectangle

Rhombus

Trapezoid

Correct Answer: D

Solution:

Based on the coordinates, the sides are not equal, and the diagonals are not equal, indicating that the quadrilateral is a trapezoid.

Yes, ABCD forms a square.

No, ABCD forms a rectangle.

No, ABCD forms a parallelogram.

No, ABCD does not form any specific quadrilateral.

Correct Answer: D

Solution:

Using the distance formula, the sides and diagonals of the quadrilateral do not satisfy the properties of a square. Therefore, ABCD does not form any specific quadrilateral.

Correct Answer: A

Solution:

Using the distance formula, the distance between points A and B is calculated as

\sqrt{(5-2)^2 + (8-4)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Correct Answer: B

Solution:

The distance between two points (x1, y1) and (x2, y2) is calculated using the formula:

\sqrt{(x2 - x1)^2 + (y2 - y1)^2}

. Substituting the given points, the distance is

\sqrt{(6 - 3)^2 + (4 - 1)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}

, which is approximately 5.

5 m

10 m

15 m

20 m

Correct Answer: B

Solution:

Since both flags are placed at the midpoints of their respective lines, and AD and BC are parallel and equal in length, the distance between the midpoints of AD and BC is equal to the length of the lines, 10 m.

6 meters

8 meters

10 meters

12 meters

Correct Answer: A

Solution:

The distance between the two flags is simply the difference in their positions along the line

AD

. Thus, it is

8 - 2 = 6

meters.

It is a rectangle.

It is a square.

It is a parallelogram.

It is a trapezium.

Correct Answer: D

Solution:

To determine the type of quadrilateral, calculate the distances:

AB = \sqrt{(5-2)^2 + (8-4)^2} = 5

BC = \sqrt{(9-5)^2 + (4-8)^2} = 5

CD = \sqrt{(9-6)^2 + (4-1)^2} = 5

DA = \sqrt{(6-2)^2 + (1-4)^2} = 5

. The diagonals

AC = \sqrt{(9-2)^2 + (4-4)^2} = 7

and

BD = \sqrt{(5-6)^2 + (8-1)^2} = 7

. Since opposite sides are equal but diagonals are not equal,

ABCD

is a trapezium.

Decrease in urban population

Increase in agricultural employment

Rise of the middle class

Decline in technological innovation

Correct Answer: C

Solution:

The Industrial Revolution led to the rise of the middle class as industries expanded, creating new job opportunities and increasing the wealth of those who owned or managed industrial enterprises. This socio-economic shift was significant in shaping modern society.

6 meters

8 meters

10 meters

12 meters

Correct Answer: A

Solution:

The distance between the two flags is the difference in their line positions, which is 8 - 2 = 6 meters.

Square

Rectangle

Rhombus

Trapezium

Correct Answer: D

Solution:

The quadrilateral formed is a trapezium because only one pair of opposite sides is parallel.

3 units

5 units

8 units

10 units

Correct Answer: A

Solution:

The vertical distance between points O and C is the difference in their y-coordinates, which is 3 units.

5 meters

6 meters

7 meters

8 meters

Correct Answer: B

Solution:

Since the flags are posted on the 2nd and 8th lines, the distance between them is 8 - 2 = 6 meters.

O(n)

O(\log n)

O(n^2)

O(1)

Correct Answer: B

Solution:

In a balanced binary search tree, the average time complexity for searching an element is

O(\log n)

, where

n

is the number of nodes in the tree. This is because the tree is traversed level by level, reducing the search space by half at each step.

Correct Answer: C

Solution:

Using the distance formula, the distance between points C and D is

\sqrt{(9-6)^2 + (4-1)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}

, which is approximately 4.24. However, the closest integer option is 5.

Square

Rectangle

Trapezoid

None of the above

Correct Answer: D

Solution:

Calculating the distances between the points, the sides do not match any specific quadrilateral properties.

36 km

15 km

39 km

51 km

Correct Answer: C

Solution:

The hypotenuse

AC

can be calculated using the Pythagorean theorem:

AC = \sqrt{AB^2 + BC^2} = \sqrt{36^2 + 15^2} = \sqrt{1296 + 225} = \sqrt{1521} = 39

km.

A character dreams of a storm before a major conflict arises.

The protagonist reflects on past events.

A detailed description of the setting.

A dialogue between two characters.

Correct Answer: A

Solution:

Foreshadowing is a literary device used to give an indication or hint of what is to come later in the story. A character dreaming of a storm before a major conflict arises is an example of foreshadowing, as it hints at future turmoil.

5 meters

6 meters

7 meters

8 meters

Correct Answer: B

Solution:

The distance between the two flags is calculated by the difference in their line positions:

8 - 2 = 6

meters.

25 km

30 km

35 km

40 km

Correct Answer: A

Solution:

Using the Pythagorean theorem:

c = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25

km.

Square

Rectangle

Rhombus

Trapezoid

Correct Answer: B

Solution:

To determine the type of quadrilateral, we calculate the distances:

AB = \sqrt{(5-2)^2 + (8-4)^2} = 5

BC = \sqrt{(9-5)^2 + (4-8)^2} = 5

CD = \sqrt{(9-6)^2 + (4-1)^2} = 5

DA = \sqrt{(6-2)^2 + (1-4)^2} = 5

. Since opposite sides are equal, it is a rectangle.

Square

Rectangle

Rhombus

Trapezium

Correct Answer: D

Solution:

Using the distance formula, the sides are AB =

\sqrt{25}

, BC =

\sqrt{25}

, CD =

\sqrt{34}

, DA =

\sqrt{34}

. The diagonals are AC =

\sqrt{68}

and BD =

\sqrt{68}

. Since opposite sides are not equal, the quadrilateral is a trapezium.

5 units

6 units

7 units

8 units

Correct Answer: A

Solution:

Using the distance formula, the distance between points A (2, 4) and B (5, 8) is calculated as:

\sqrt{(5-2)^2 + (8-4)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

units.

5 m

6 m

7 m

8 m

Correct Answer: B

Solution:

The distance between the flags is the difference in their positions: 8 - 2 = 6 m.

(2.5, 4)

(2, 3)

(3, 4)

(3.5, 5)

Correct Answer: A

Solution:

The midpoint is calculated as:

\left( \frac{1+4}{2}, \frac{2+6}{2} \right) = (2.5, 4)

18 square units

24 square units

30 square units

36 square units

Correct Answer: A

Solution:

To find the area of quadrilateral

ABCD

, we can divide it into two triangles:

\triangle ABD

and

\triangle BCD

. Using the formula for the area of a triangle given vertices

(x_1, y_1)

(x_2, y_2)

, and

(x_3, y_3)

\text{Area} = \frac{1}{2} | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) |

. For

\triangle ABD

\text{Area} = \frac{1}{2} | 3(4-6) + 6(6-1) + 3(1-4) | = 9

. For

\triangle BCD

\text{Area} = \frac{1}{2} | 6(6-6) + 9(6-4) + 3(4-6) | = 9

. Total area =

9 + 9 = 18

square units.

True or False

Correct Answer: True

Solution:

The grid is described as having horizontal and vertical lines creating a 10x10 matrix, with both axes labeled from 0 to 10.

Correct Answer: True

Solution:

Solution:

Using the distance formula, the distance between points A (2, 4) and C (9, 4) is calculated as

Solution:

Using the distance formula, the distances between the points are not equal, hence they do not form a square.

Correct Answer: False

Solution:

The distance between points A(2, 4) and C(9, 4) is calculated using the distance formula:

\sqrt{(9-2)^2 + (4-4)^2} = \sqrt{49} = 7

. However, the correct distance given in the excerpt is 68 units, indicating a misunderstanding or error in the excerpt.

Correct Answer: True

Solution:

According to the diagram description, point D is indeed located at (6, 1) on the grid.

Correct Answer: True

Solution:

The excerpt describes the grid as having 10 columns and 10 rows, confirming its dimensions.

Coordinate Geometry

AI Learning Assistant

Summary

Chapter Summary

Key Points

Examples

Applications

Learning Objectives

Detailed Notes

Chapter 7: Coordinate Geometry

7.1 Introduction

7.2 Distance Formula

Examples

7.3 Section Formula

Examples

Exercises

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.In a classroom, four friends are seated at points A(2, 4), B(5, 8), C(9, 4), and D(6, 1). Using the distance formula, determine if the quadrilateral ABCD is a square.

Correct Answer: B

Q2.In a rectangular school ground ABCD, lines are drawn with chalk powder at a distance of 1m each. If Niharika runs along line AD and posts a green flag at the 4th line, and Preet runs along the 8th line and posts a red flag, what is the distance between the two flags?

Correct Answer: A

Q3.If point A is located at (2, 4) and point B is located at (5, 8), what is the distance between these two points?

Correct Answer: A

Q4.In a coordinate plane, if point PPP is at (3,5)(3, 5)(3,5) and point QQQ is at (7,9)(7, 9)(7,9), what is the midpoint of the line segment PQPQPQ?

Correct Answer: A

Q5.A chemical reaction is given by 2A+B→C+D2A + B \rightarrow C + D2A+B→C+D. If the initial concentration of AAA is 0.5 M and BBB is 0.3 M, and at equilibrium, the concentration of CCC is 0.2 M, what is the equilibrium concentration of AAA?

Correct Answer: B

Q6.If a right triangle has one leg measuring 36 km and the other leg measuring 15 km, what is the length of the hypotenuse?

Correct Answer: B

Q7.In a coordinate plane, points A, B, C, and D are located at (2, 4), (5, 8), (9, 4), and (6, 1) respectively. Calculate the distance between points A and C.

Correct Answer: B

Q8.In the coordinate system of Fig. 7.1, if point A is at the origin and point B is 36 km east, what is the distance from point A to point B?

Correct Answer: B

Q9.What is the length of the line segment joining points A(2, 4) and B(5, 8)?

Correct Answer: D

Q10.In a coordinate plane, what is the slope of the line passing through points B(5, 8) and D(6, 1)?

Correct Answer: A

Q11.Given the points A(2, 4), B(5, 8), C(9, 4), and D(6, 1) on a coordinate plane, which of the following correctly describes the type of quadrilateral formed by these points?

Correct Answer: A

Q12.In the coordinate plane, what is the distance between points A(2, 4) and C(9, 4)?

Correct Answer: B

Q13.What is the type of quadrilateral formed by points A(2, 4), B(5, 8), C(9, 4), and D(6, 1)?

Correct Answer: D

Q14.If a right triangle has legs of lengths 9 units and 12 units, what is the length of the hypotenuse?

Correct Answer: A

Q15.In a coordinate plane, if point A is at (2, 4) and point D is at (6, 1), what is the slope of the line passing through these points?

Correct Answer: D

Q16.In a chemical reaction, 2 moles of hydrogen gas react with 1 mole of oxygen gas to form water. How many moles of water are produced?

Correct Answer: B

Q17.In the coordinate plane, what is the midpoint of the line segment joining points A(2, 4) and C(9, 4)?

Correct Answer: A

Q18.If point A is located at (2, 4) and point C is located at (9, 4), what is the horizontal distance between these points?

Correct Answer: A

Q19.If a point P is located at (3, 1) and another point Q is located at (6, 4), what is the midpoint of the line segment PQ?

Correct Answer: A

Q20.In a coordinate plane, points A(2, 4), B(5, 8), C(9, 4), and D(6, 1) form a quadrilateral. Using the distance formula, which of the following best describes the type of quadrilateral ABCD?

Correct Answer: D

Q21.If point A is located at (2, 4) and point D is located at (6, 1), what is the distance between these two points?

Correct Answer: A

Q22.In a binary tree, each node has either 0 or 2 children. If there are 15 leaf nodes, how many total nodes are there in the tree?

Correct Answer: C

Q23.In the coordinate plane, if point A is at (0, 0) and point B is at (6, 8), what is the distance between points A and B?

Correct Answer: A

Q24.In a hypothetical ecosystem, if the primary producers produce 1000 kcal of energy, how much energy is available to the tertiary consumers assuming 10% energy transfer efficiency at each trophic level?

Correct Answer: B

Q25.What is the length of the line segment joining points C(9, 4) and D(6, 1)?

Correct Answer: D

Q26.Consider a hypothetical scenario where a new species of plant has evolved to have leaves that can photosynthesize at night using a novel enzyme. What evolutionary advantage might this provide?

Correct Answer: B

Q27.What is the slope of the line passing through points A(2, 4) and B(5, 8)?

Correct Answer: A

Q28.In a classroom, 4 friends are seated at points A, B, C, and D. Champa thinks ABCD is a square, while Chameli disagrees. Using the distance formula, who is correct?

Correct Answer: B

Q29.In a grid, if point B is at (5, 8) and point D is at (6, 1), what is the vertical distance between these points?

Correct Answer: B