Class 10 RD Sharma Solutions — Глава 14 Координатная геометрия — Упражнение 14.3 | Набор 2

Вопрос 21. Найдите отношение, в котором точка P(-1, y), лежащая на отрезке, соединяющем A(-3, 10) и B(6, -8), делит его. Кроме того, найдите значение y.

Решение:

Assume P divide A(-3, 10) and B(6, -8) in the ratio of k : 1

Given: coordinates of P as (-1, y)

After applying the section formula for x – coordinate,

We will get

Therefore, 

AB is divided by point P in the ratio of 2 : 7

By applying the value of k, to find the y-coordinate 

We will get

y = (-16 + 70)/(2 + 7) = 54/9

y = 6 

Hence, 

The y-coordinate of P is 6.

Вопрос 22. Найдите координаты точки A, где AB — диаметр окружности, центр которой равен (2, -3), а B равен (1, 4).

Решение:

Assume the coordinates of point A be (x, y)

Given: AB is the diameter, 

So the center in the mid-point of the diameter

Thus,

(2, -3) = (x + 1/ 2, y + 4/2)

2 = x + 1/2 and -3 = y + 4/2 

4 = x + 1 and -6 = y + 4

x = 3 and y = -10

Hence, the coordinates of A are (3, -10)

Вопрос 23. Если точки (-2, 1), (1, 0), (x, 3) и (1, y) образуют параллелограмм, найти значения x и y.

Решение:

Consider A(-2, 1), B(1, 0), C(x , 3) and D(1, y) are the given points of the parallelogram.

As we know that the diagonals of a parallelogram bisect each other.

Thus, 

Coordinates of mid-point of AC = Coordinates of mid-point of BD

((x – 2)/2, (3 – 1)/2) = (1+1)/2, (y + 0)/2 

((x – 2)/2, 1) = (1, y/2) 

(x – 2)/2 = 1

x – 2 = 2

x = 4

and y/2 = 1

y = 2

Hence, the value of x is 4 and the value of y is 2.

Вопрос 24. Точки A(2, 0), B(9, 1), C(11, 6) и D(4, 4) являются вершинами четырехугольника ABCD. Определить, является ли ABCD ромбом или нет.

Решение:

Given: A(2, 0), B(9, 1), C(11, 6) and D(4, 4).

Mid-point of AC coordinates are 

Mid-point of BD coordinates are 

Here, 

Coordinates of the mid-point of AC ≠ Coordinates of mid-point of BD, 

ABCD is not a parallelogram.

Hence, 

ABCD cannot be a rhombus too.

Вопрос 25. В каком отношении точка (-4,6) делит отрезок, соединяющий точки A(-6,10) и B(3,-8)?

Решение:

Assume the line segment AB is divided by point (-4, 6) in the ratio of k : 1.

After applying the Section Formula, 

We will get

-4k -4 = 3k – 6

7k = 2

k : 1 = 2 : 7

We can also check for the y-coordinate also.

Hence, 

The ratio in which the line segment AB is divided by point (-4,6) is 2 : 7.

Вопрос 26. Найдите отношение, в котором ось у делит отрезок, соединяющий точки (5, -6) и (-1, -4). Также найдите координаты точки разделения.

Решение:

Assume P(5, -6) and Q(-1, -4) be the given points.

Consider the line segment PQ is divided by y-axis in the ratio k : 1.

After applying the Section Formula for the x-coordinate (as it’s zero) 

We will get,

-k + 5 = 0

k = 5

Therefore, 

The ratio in which the y-axis divides the given 2 points is 5 : 1

Now further, for finding the coordinates of the point of division

On putting k = 5, we will get

Therefore, 

The coordinates of the point of division are (0, -13/3) 

Вопрос 27. Докажите, что A(-3, 2), B(-5, 5), C(2, -3) и D(4, 4) — вершины ромба.

Решение:

Given: A(-3, 2), B(-5, 5), C(2, -3) and D(4, 4)

Further,

Mid-point of AC coordinates are 

And,

Mid-point of BD coordinates are 

Therefore, 

The mid-point for both the diagonals are the same. 

Thus, 

ABCD is a parallelogram.

Now, 

For the sides

AB = BC

We can see that ABCD is a parallelogram with adjacent sides equal.

Therefore, 

ABCD is a rhombus.

Вопрос 28. Найдите длины медиан ΔABC с вершинами в точках A(0,-1), B(2,1) и C(0,3).

Решение:

Assume AD, BE and CF be the medians of ΔABC

Now,

Coordinates of D are  = (1, 2)

Coordinates of E are  = (0, 1)

Coordinates of F are  = (1, 0)

Further,

The length of the medians

Length of the median AD =  = √10 units

Length of the median BE =  = 2 units

Length of the median CF =  = = √10 units

Вопрос 29. Найдите длину медианы ΔABC с вершинами в A(5, 1), B(1, 5) и C(-3, -1).

Решение:

Given: Vertices of ΔABC as A(5, 1), B(1, 5) and C(-3, -1).

Consider AD, BE and CF be the medians

Coordinates of D are  = (-1, 2)

Coordinates of E are  = (1, 0)

Coordinates of F are  = (3, 3)

Further, 

The length of the medians

Length of the median AD =  = √37 units

Length of the median BE =  = 5 units

Length of the median CF =  = √52 units

Вопрос 30. Найдите координаты точки, которая делит отрезок, соединяющий точки (-4, 0) и (0, 6), на четыре равные части.

Решение:

Consider A(-4, 0) and B(0, 6) as they are the given points

And, 

Assume P, Q and R be the points which divide AB is four equal points, as shown in the fig.

Thus, 

As we know that AP : PB = 1 : 3

By applying the Section Formula the coordinates of P are

And,

We can see that Q is the mid-point of AB

Thus, the coordinates of Q are

Finally, 

The ratio of AR : BR is 3 : 1

Then, after applying the Section Formula the coordinates of R are

Вопрос 31. Покажите, что середина отрезка, соединяющего точки (5, 7) и (3, 9), является также серединой отрезка, соединяющего точки (8, 6) и (0, 10). ).

Решение:

Assume M be the mid-point of AB. Coordinates of the mid-point of this line segment joining two points A (5, 7) and B (3, 9).

Now coordinates of the mid-point of the line segment joining the points (8, 6) and (0, 10) are;

Thus, this is the same as the first case.

Вопрос 32. Найдите расстояние от точки (1, 2) до середины отрезка, соединяющего точки (6, 8) и (2, 4).

Решение:

Assume M be the mid-point of the line segment joining the points (6, 8) and (2, 4)

Now 

Coordinates of M will be

Now, 

Distance between the points (4, 6) and (1, 2)

 = 5 units

Вопрос 33. Если A и B равны (1, 4) и (5, 2) соответственно, найти координаты точки P. Когда

Решение:

Here, Point P divides the line segment joining the points (1, 4) and (5, 2) in the ratio of AP : PB = 3 : 4

Coordinates of P will be

Вопрос 34. Докажите, что точки A (1, 0), B (5, 3), C (2, 7) и D (-2, 4) являются вершинами параллелограмма.

Решение:

If ABCD is a parallelogram, 

Then its diagonal AC and BD will bisect each other at O

Consider O is the mid-point of AC, 

Then coordinates of O will be;

And assume O is the mid-point of BD,

Then coordinates of O will be;

We see that coordinates of the mid-points of AC and BD are same

Therefore, AC and BD  bisect each other at O

Now, length of AC 

and length of BD = 

We can see that AC = BD

Therefore, ABCD is a rectangle.

Вопрос 35. Определите отношение, в котором точка P (m, 6) делит соединение A (-4, 3) и B (2, 8). Кроме того , найдите значение m.

Решение:

Assume the ratio be r : s in which P (m, 6) divides the line segment joining the points A (-4, 3) and B (2, 8)

Therefore,

and 

⇒ 8r + 3s = 6r + 6s

⇒ 8r – 6r = 6s – 3s

⇒ 2r = 3s

Therefore,

Ratio is 3 : 2

Now, 

Hence, m = -2/5 

Вопрос 36. Определите отношение, в котором точка (-6, а) делит соединение А (-3, -1) и В (-8, 9). Кроме того , найдите значение a.

Решение:

Assume the point P (-6, a) divides the join of A (-3, -1) and B (-8, 9) in the ratio m : n

Therefore,

-6 = (-8m -3n)/(m + n)

-6m – 6n = -8m – 3n

8m – 6m = 6n – 3n

2m = 3n

m/n = 3/2

Therefore,

Ratio = 3 : 2

and

Вопрос 37. ABCD — прямоугольник, образованный соединением точек A (-1, -1), B (-1, 4), C (5, 4) и D (5, -1). P, Q, R и S — середины сторон AB, BC, CD и DA соответственно. Является ли четырехугольник PQRS квадратом? Прямоугольник ? или ромб? Обосновать ответ.

Решение:

ABCD is a rectangle whose vertices are A (-1,-1), B (-1,4), C (5, 4) and D (5, -1). 

P, Q, R, and S are the mid-points of the sides AB, BC, CD and DA respectively 

And are joined PR and QS are also joined.

Now coordinates of P will be

Similarly, the coordinates of Q, will be:

Coordinates of R will be:

Coordinates of S will be:

Coordinates of P (-1, 3/2), Q (2, 4), R(5, 3/2) and S (2, -1)

Now, assume the diagonals PQ and QS intersect each other at O

Assume O is the mid-point of PR,

Then coordinates of O will be

Similarly, of O is the mid-point of QS, then the coordinates of O will be

Now, we see that the coordinates of O in both case is same and adjacent sides are also equal

Then it may be a square or a rhombus

Now length of PR = 

And length of OS

Because diagonal are not equal

Hence, PQRS is a rhombus.

Вопрос 38. Точки P, Q, R и S делят отрезок, соединяющий точки A (1, 2) и B (6, 7), на 5 равных частей. Найдите координаты точек P, Q и R.

Решение:

Points P, Q, R and S divides AB in 5 equal parts and assume coordinates of P, Q, R and S are,

(x₁, y₁), (x₂, y₂), (x₃, y₃), (x₄, y₄) 

⇒ P divides AB in ratio 1 : 4

Therefore,

Hence, Coordinates of P are (2, 3)

⇒ Q divides AB in the ratio 2 : 3

Therefore,

Hence, Coordinates of 3, 4

⇒ R divides AB in ration 3 : 2

Therefore,

Hence, Coordinates of R are (4, 5).

Вопрос 39. Если A и B — две точки с координатами (-2, -2) и (2, -4) соответственно, найдите координаты P такие, что AP = 3/7 AB

Решение:

AP = 3/7 AB

7AP = 3AB

7AP = 3(AP + BP)

⇒ 7AP = 3AP + 3BP

⇒ 7AP – 3AP = 3BP

⇒ 4 AP = 3 BP

⇒ 

Therefore, 

AP : BP = 3 : 4

Because P divides AB in the ratio of 3 : 4 whose end points are A(-2, -2) and B(2, -4)

Therefore, Coordinates of P will be

Therefore,

Coordinates of P will be 

Вопрос 40. Найдите координаты точек, которые делят отрезок, соединяющий A (-2, 2) и B (2, 8), на четыре равные части.

Решение:

Assume P, Q and R divides the line segment AB in four equal parts

Co-ordinates of A are (-2, 2) and of B are (2, 8)

It can be seen that Q divides AB in two equal parts while P bisects AQ and R, bisect QB.

Now,

Coordinates of Q will be :

Similarly, coordinates of P will be:

Coordinates of R will be:

Hence, Coordinates of P are(1, 7/2) 

Coordinates of Q are (0, 5)

Coordinates of R are (1, 13/2)