Question

The distance between the points \((3 \mathrm{k}+1,-3 \mathrm{k})\) and \((3 \mathrm{k}-2,-4-3 \mathrm{k})\) (in units) is (1) \(3 \mathrm{k}\) (2) \(5 \mathrm{k}\) (3) 5 (4) 3

Short answer

Answer: The distance between the two points is 5 units.

Step-by-step solution

Identify the coordinates of the given points

The coordinates of the points given are: Point 1, P1: \((3k+1, -3k)\) Point 2, P2: \((3k-2, -4-3k)\)

Apply the distance formula

In order to find the distance between the two points, we will use the distance formula \(D=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\): \(D = \sqrt{[(3k - 2) - (3k + 1)]^2 + [(-4 -3k) - (-3k)]^2}\)

Simplify the expression

We need to expand and simplify the terms inside the square root: \(D = \sqrt{[(3k - 2 - 3k - 1)]^2 + [(-4 - 3k + 3k)]^2}\) \(D = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

Identify the correct answer

The result of calculating the distance between the two points is D = 5. This corresponds to option (3). So, the correct answer is 5 units.

Key concepts

Coordinate Geometry

Coordinate Geometry is a fundamental part of mathematics that deals with points, lines, and shapes in a coordinate plane. It allows us to calculate positions using numerical coordinates and perform geometric tasks using algebraic methods. In this concept, each point on the plane is expressed by a pair of coordinates. For instance, the point given by \((3k+1, -3k)\) means the x-coordinate is \(3k+1\) and the y-coordinate is \(-3k\). Understanding how to interpret and manipulate these coordinates is key in solving geometry problems.

Coordinate Geometry integrates both algebra and geometry, making it a powerful tool in many fields of mathematics and science. Key elements like the distance formula and section formula help solve real-world problems, from determining distances between GPS coordinates to mapping out strategic points in engineering.

Distance Calculation

The Distance Calculation between two points is essential for determining the direct line length that connects them in a coordinate plane. In our exercise, we use the formula \[D = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\], where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points.

This formula is derived from the Pythagorean theorem, which is used to find the hypotenuse of a right triangle formed by the differences in x and y coordinates. By identifying the changes in the x-direction and the y-direction, we form a triangle and calculate the length of the hypotenuse between the two points.

• Step 1: Determine the differences in x-coordinates and y-coordinates.
• Step 2: Square each difference.
• Step 3: Add the squared differences.
• Step 4: Take the square root of this sum to find the distance.

In practical applications, distance calculations assist in physics for calculating force vectors, in navigation systems for route mapping, and in various engineering disciplines.

Mathematics Problem Solving

Mathematics Problem Solving involves using logical steps to find solutions to given problems. Problems in mathematics often require applying various concepts together, as seen in our problem with distance calculation. Here, we carefully follow a set method to reach the answer.

To effectively solve such a problem:

• First, identify known elements and choose the proper formula or technique to apply, such as the distance formula for calculating distance between points.
• Next, substitute in known values and simplify each expression step by step, ensuring accuracy along the way.
• After calculation, review and verify that the results align logically with the context of the problem.

This structured approach enhances understanding and solution efficiency. Mathematics Problem Solving not only equips students to handle various mathematical contexts but also fosters critical thinking and analytical skills applicable beyond class.

Class 10 Mathematics

Class 10 Mathematics forms a critical academic foundation, focusing on a variety of concepts, including Coordinate Geometry and problem-solving techniques like Distance Calculation. At this level, students are introduced to a range of mathematical ideas that are crucial for higher education and practical applications.

Key areas covered include basics of algebra, trigonometry, geometry, and statistics. By building a robust understanding of the fundamentals through exercises like our distance problem, students prepare themselves for challenges in advanced mathematics courses.

Educators emphasize practice through problems to solidify concepts and nurture confidence in problem-solving abilities. As mathematics is a progressive subject, mastering concepts in Class 10 Maths ensures future academic success and cultivates an appreciation for the analytical nature of mathematical reasoning.