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The fundamental understanding of point relationships within coordinate spaces is vital while studying geometry. Points positioned on one straight line are labelled as collinear points when there are three or more. The detection of collinearity requires using both distance formula computations and slope comparisons between segments that connect the points. The diagnostic tools enable verification of point position in graphs, which ultimately supports both proof building and analytical approaches. The development of this skill leads to better preparation when solving coordinate geometry problems.
This section of the NCERT Solutions for Class 10 Maths emphasises the practical application of the distance formula. The exercises require students to establish point-to-point distance measurements, along with confirming points lie on the same straight line and identifying various triangle and quadrilateral shapes through their side lengths. Students improve their mastery of coordinate geometry through the practice of Exercise 7.1 from the NCERT Books and develop critical analytical skills that will serve them in advanced mathematical learning.
Coordinate Geometry Class 10 Chapter 7 Exercise: 7.1
Q1 (i) Find the distance between the following pairs of points : (2, 3), (4, 1)
Answer:
Given points: (2, 3), (4, 1)
Distance for the points
After putting values, we get:
Q1 (ii) Find the distance between the following pairs of points : (– 5, 7), (– 1, 3)
Answer:
Given points: (– 5, 7), (– 1, 3)
Distance for the points
After putting values, we get:
Q1 (iii) Find the distance between the following pairs of points :(a, b), (– a, – b)
Answer:
Given points: (a, b), (– a, – b)
Distance for the points
After putting values, we get:
Answer:
Given points: (0, 0) and (36, 15)
Distance for the points
After putting values we get:
Therefore, the distance between the two towns A and B is, thus, 39 km for the given towns' location
Q3 Determine if the points (1, 5), (2, 3), and (– 2, – 11) are collinear.
Answer:
Let the points (1, 5), (2, 3), and (– 2, – 11) represent the vertices A, B, and C of the given triangle, respectively.
To determine whether these points are collinear or not, we need to verify if the sum of the lengths of any two line segments is equal to the length of the third line segment.
Therefore,
Thus, the cases do not satisfy the condition for collinear points
Hence, the points are not collinear.
Q4. Check whether (5, – 2), (6, 4) and (7, – 2) are the vertices of an isosceles triangle.
Answer:
We know the isosceles triangle has two equal sides. So, we will check the distances between the points to verify whether it is an isosceles triangle or not.
The distance between two points
So, we have the following points: (5, – 2), (6, 4) and (7, – 2), assuming them to be the vertices of triangle A, B, and C, respectively.
Therefore, AB = BC
Here two sides are equal in length.
Therefore, ABC is an isosceles triangle.
Answer:
The coordinates of the points:
The distance between two points
Hence,
And the lengths of diagonals:
So, here it can be seen that all sides of the quadrilateral ABCD are of the same length and the diagonals are also having the same length.
Therefore, quadrilateral ABCD is a square, and Champa is right.
Answer:
Let the given points
Apply the distance formula:
After putting in the values, we get:
Finding the length of the diagonals:
Thus, it is clear that all sides are of the same length, and also the diagonals have the same length.
Hence, the given quadrilateral is a square.
Q6 (ii) Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer: (–3, 5), (3, 1), (0, 3), (–1, – 4)
Answer:
Let the given points
Apply the distance formula:
After putting in the values, we get:
All the sides of the given quadrilateral have different lengths.
Therefore, it is only a general quadrilateral and not a specific one like a square, rectangle, etc.
Q6 (iii) Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer: (4, 5), (7, 6), (4, 3), (1, 2)
Answer:
Let the given points
Apply the distance formula:
After putting in the values, we get:
And the diagonals:
Here we can observe that the opposite sides of this quadrilateral are of the same length.
However, the diagonals are of different lengths.
Therefore, the given points are the vertices of a parallelogram.
Q7 Find the point on the x-axis which is equidistant from (2, -5) and (–2, 9).
Answer:
Let the point which is equidistant from
Then, we have
Distance AX:
According to the question, these distances are of equal length.
Hence, we have,
Solving this to get the required coordinates.
Squaring both sides, we get,
Or,
Hence, the point is
Q8 Find the values of y for which the distance between the points P(2, – 3) and Q(10, y) is 10 units.
Answer:
Given the distance between the points
The distance formula :
So, given
After squaring both sides
Therefore, the values are
Answer:
Given
Then, the distances
Distance
Distance
Squaring both sides, we get
The points are:
CASE I: when R is
The distances QR and PR.
CASE II: when R is
The distances QR and PR.
Answer:
Let the point
Then, the distances
Squaring both sides: we obtain
Thus, the relation is
Also Read-
1. Distance Formula: The distance formula enables users to find the distance between points by analysing their position through coordinates.
2. Collinearity of Points: The evaluation of straight-line coexistence for three or more points represents Collinearity of Points.
3. Classification of Triangles: The identification process for triangles focuses on determining their three types (equilateral, isosceles, or scalene) through analysing side length measurements.
4. Classification of Quadrilaterals: The classification of quadrilaterals detects the specific shape among square, rectangle, rhombus and other four-point formed shapes.
5. Real-Life Applications: The distance formula finds use in solving practical distance problems that emerge throughout different settings.
Check Out-
Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.
As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters
Students must check the NCERT Exemplar solutions for class 10 of Mathematics and Science Subjects.
We assume that the line between two coordinates is hypotenuse so with the help x axis and y axis we will find perpendicular and base and by applying pythagoras theorem we will find hypotenuse .
H =square root of [(P)²+(B)²]
In the above formula, H is the hypotenuse, P is perpendicular and B is base
Collinear points are set of three or more points which is on exact same line
The distance formula is used in NCERT solutions for Class 10 Maths exercise 7.1 to find area of triangle length of vertices of different types of triangles such as an isosceles triangle.
Set of points all at equal distance from each other.
Before the Class 10 Mathematics chapter 7 activity 7.1, there are five key questions that must be answered.
There are 10 questions in Class 10 Maths chapter exercise 7.1. Question 1 consists of three subparts.
Question 1 and question 2 is very basic coordinates are given and we have to find the distance then we have question 3 based on collinear point question 4 on finding vertices of an isosceles triangle then we have question 5 based on real-life scenario and then we have some question of equidistant point
Admit Card Date:17 April,2025 - 17 May,2025
Exam Date:01 May,2025 - 08 May,2025
Hello
Since you are a domicile of Karnataka and have studied under the Karnataka State Board for 11th and 12th , you are eligible for Karnataka State Quota for admission to various colleges in the state.
1. KCET (Karnataka Common Entrance Test): You must appear for the KCET exam, which is required for admission to undergraduate professional courses like engineering, medical, and other streams. Your exam score and rank will determine your eligibility for counseling.
2. Minority Income under 5 Lakh : If you are from a minority community and your family's income is below 5 lakh, you may be eligible for fee concessions or other benefits depending on the specific institution. Some colleges offer reservations or other advantages for students in this category.
3. Counseling and Seat Allocation:
After the KCET exam, you will need to participate in online counseling.
You need to select your preferred colleges and courses.
Seat allocation will be based on your rank , the availability of seats in your chosen colleges and your preferences.
4. Required Documents :
Domicile Certificate (proof that you are a resident of Karnataka).
Income Certificate (for minority category benefits).
Marksheets (11th and 12th from the Karnataka State Board).
KCET Admit Card and Scorecard.
This process will allow you to secure a seat based on your KCET performance and your category .
check link for more details
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Hope this helps you .
Hello Aspirant, Hope your doing great, your question was incomplete and regarding what exam your asking.
Yes, scoring above 80% in ICSE Class 10 exams typically meets the requirements to get into the Commerce stream in Class 11th under the CBSE board . Admission criteria can vary between schools, so it is advisable to check the specific requirements of the intended CBSE school. Generally, a good academic record with a score above 80% in ICSE 10th result is considered strong for such transitions.
hello Zaid,
Yes, you can apply for 12th grade as a private candidate .You will need to follow the registration process and fulfill the eligibility criteria set by CBSE for private candidates.If you haven't given the 11th grade exam ,you would be able to appear for the 12th exam directly without having passed 11th grade. you will need to give certain tests in the school you are getting addmission to prove your eligibilty.
best of luck!
According to cbse norms candidates who have completed class 10th, class 11th, have a gap year or have failed class 12th can appear for admission in 12th class.for admission in cbse board you need to clear your 11th class first and you must have studied from CBSE board or any other recognized and equivalent board/school.
You are not eligible for cbse board but you can still do 12th from nios which allow candidates to take admission in 12th class as a private student without completing 11th.
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