if A=(1,0),B=(7,0)C=(9,3)D=(3,3)
then ABCD is
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Answer:
Exercise 7.1
Q.1. Find the distance between the following pairs of points:
(i) (2, 3), (4, 1) (ii) (–5, 7), (–1, 3) (iii) (a, b), (–a, –b)
Sol. (i) Here x1 = 2, y1 = 3, x2 = 4 and y2 = 1
∴ The required distance
Q.2. Find the distance between the point (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2 of the NCERT textbook?
Sol. Part-I
Let the points be P(0, 0) and Q(36, 15).
Q.3. Determine if the points (1, 5), (2, 3) and (�2, �11) are collinear.
Sol. Let the points be A(1, 5), B (2, 3) and (�2, �11)
A, B and C are collinear, if
AB + BC = AC
AC + CB = AB
But AB + BC ≠ AC
AC + CD ≠ AB
BA + AC ≠ BC
∴ A, B and C are not collinear.
Q.4. Check whether (5, –2), (6, 4) and (7, –2) are the vertices of an isosceles triangle.
Sol. Let the points be A (5, –2), B (6, 4) and (7, –2).
We have AB = BC ≠ AC
∴ΔABC is an isosceles triangle.
Q.5. In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. Champa and Chaneli walk into the class and after observing for a few minutes Champa asks Chaneli, “Don�t you think ABCD is a squar?” Chaneli disagrees. Using distance single formula, find which of them is correct.
Sol. Let the number of horizontal columns represent the x-coordinates whereas the vertical rows represent the y-coordinates.
∴ The points are:
A(3, 4), B(6, 7), C(9, 4) and D(6, 1)
Q.6. Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(i) (�1, �2), (1, 0), (�1, 2), (�3, 0)
(ii) (�3, 5), (3, 1), (0, 3), (�1, �4)
(iii) (4, 5), (7, 6), (4, 3), (1, 2)
Sol. (i) Let the points be: A(�1, �2), B(1, 0), C(�1, 2) and D(�3, 0).
(ii) Let the points be A (�3, 5), B(3, 1), C(0, 3) and D(�1, �4).
⇒ A, B, C and D are collinear. Thus, ABCD is not a quadrilateral.
(iii) Let the points be A (4, 5), B (7, 6), C (4, 3) and D(1, 2).
Q.7. Find the point on the x-axis which is equidistant from (2, �5) and (�2, 9).
Sol. We know that any point on x-axis has its ordinate = 0.
Let the required point be P(x, 0).
Let the given points be A(2, �5) and B(�2, 9)
Q.8. Find the values of y for which the distance between the points P(2, �3) and Q(10, y) is 10 units.
Sol. The given points are P(2, �3) and Q(10, y).
Q.9. If Q (0, 1) is equidistant from P(5, �3) and R(x, 6), find the values of x. Also find the distances QR and PR.
Q.10. Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (�3, 4).
Sol. Let the points be A(x, y), B (3, 6) and C(�3, 4).
Squaring both sides,
(3 – x)2 + (6 – y)2 = (–3 – x)2 + (4 – y)2
⇒ (9 + x2 – 6x) + (36 + y2 – 12y) = (9 + x2 + 6x) + (16 + y2 – 8y)
⇒ 9 + x2 – 6x) + 36 + y2 – 12y – 9 – x2 – 6x – 16 – y2 + 8y
⇒ –6x – 6x + 36 – 12y – 16 + 8y = 0
⇒ –12x – 4y + 20 = 0
⇒ –3x – y + 5 = 0
⇒ 3x + y – 5 = 0
which is the required relation between x and y.
Exercise 7.2
Q.1. Find the coordinates of the point which divides the join of (�V1, 7) and (4, �V3) in the ratio 2 : 3.
Sol. Let the required point be P (x, y).
Here, the end points are:
(–1, 7) and (4, –3)
∵Ratio = 2 : 3 = m1 : m2
Q.2. Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).
Sol. Let the given points be A(4, –1) and B(–2, –3).
Let the poi ts P and Q trisect AB.
i.e., AP = PQ = QB
i.e., P divides AB in the ratio of 1 : 2
Q divides AB in the ratio of 2 : 1
Let the coordinates fo P be (x, y).
Q.3. To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in the figure. Niharika runs th the distance AD on the 2nd line and posts a green flag. Preet runs th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?
Sol. Let us consider ‘A‘ as origin, then AB is the x-axis. AD is the y-axis.
Now, the position of green flag-post is
And the position of red flag-post is
or x = 5 and y = (22.5).
Thus, the blue flag is on the 5th line at a distance 22.5 m above AB.
Q.4. Find the ratio in which the line segment joining the points (–3, 10) and (6, –8) is divided by (–1, 6).
Sol. Let the given points are: A (–3, 10) and B (6, –8).
Let the point P (–1, 6) divides AB in the ratio m1 : m2.
Q.5. Find the ratio in which the line segment joining A (1, –5) and B (–4, 5) is divided by the x-axis. Also find the coordinates of the point of division.
Sol. The given points are: A (1, –5) and B (–4, 5).
Let the required ratio = k : 1 and the required point be P (x, y).
Part-I: To find the ratio
Since the point P lies on x-axis,
∴ Its y-coordinates is 0.
Step-by-step explanation:
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