# Plus One Maths Chapter Wise Questions and Answers Chapter 12 Introduction to Three Dimensional Geometry

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## Kerala Plus One Maths Chapter Wise Questions and Answers Chapter 12 Introduction to Three Dimensional Geometry

### Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers

Question 1.
Prove by using distance formula that the A(1, 2, 3), B(-1, -1, -1) and C(3, 5, 7) are collinear. Now BC = AB + AC
Thus A, B, C are collinear. Question 2.
Verify the following: (3 score each)

1. (0, 7, -10), (1, 6, -6) and (4, 9, -6) are the vertices of an isosceles triangle.
2. (0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of a right angled triangle.
3. (-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are the vertices of a parllelogram.

1. Let A(0, 7, -10), B(1, 6, -6) and C(4, 9, -6) be the two points. Now AB = BC, thus ABC is an isosceles triangle.

2. Let A(0, 7, 10), B(-1, 6, 6) and C(-4, 9, 6) be the two points.  Now AC2 = AB2 + BC2, thus ABC is a right triangle.

3. Let A(-1, 2, 1), B(1, -2, 5), C(4, -7, 8) and D(2, -3, 4) be the two points. Now AB = CD, BC = AD, AC ≠ BD, thus A, B, C, D are vertices of a parallelogram. Question 3.
Find the equation of set points which are equidistant from the points (1, 2, 3) and (3, 2, -1).
Let P(x,y,z) be any point which is equidistant from the point A(1, 2, 3) and B (3, 2, -1).
Given; PA = PB (x – 1)2 + (y – 2)2 + (z – 3)2
= (x – 3)2 + (y – 2)2 + (z + 1)2
= x2 – 2x + 1 + y2 – 4y + 4 + z2 – 6z + 9
= x2 – 6x + 9 + y2 – 4y + 4 + z2 + 2z + 1 – 2x + 14 – 6z = -6x + 14 + 2z
⇒ 4x – 8z = 0
⇒ x – 2z = 0.

Question 4.
Find the coordinate of the point which divides the line segment joining the points (3, -2, 5) and (3, 4, 2) in the ratio 2:1 (3 score each)

1. 2:1 internally
2. 2:1 externally

1. Let P(x, y, z) be any point which divides the line segment joining points A(3, -2, 5) and B (3, 4, 2) in the ratio 2:1 internally. Therefore coordinates of P are (3, 2, 3).

2. Let P(x, y, z) be any point which divides the line segment joining points A(3, -2, 5) and B (3, 4, 2) in the ratio 2:1 externally. Therefore coordinates of P are (3, 10, -1). Question 5.
Find the ratio in which the line joining the points (1, 2, 3) and (-3, 4, -5) is divided by the xy-plane.
Let the line joining the points A(1, 2, 3) and B(-3, 4, -5) is divided by the xy-plane in the ratio k:1.
Then the coordinate Since the point lies on xy-axis, we have; Thus the required ratio is $$\frac{3}{5}$$; ie, 3:5.

Question 6.
Find the coordinates of the points which trisect the line segment joining the points P(4, 2, -6) and Q (10, -16, 6). Let R and S be two points which trisect the line join of PQ. Therefore PR = RS = SQ Then coordinate of R is = (6, -4, -2)
Then coordinate of S is = (8, -10, 2).

### Plus One Maths Introduction to Three Dimensional Geometry Practice Problems Questions and Answers

Question 1.
Find the distance between the following pair of points: (1 score each)

1. (2, 3, 5) and (4, 3, 1)
2. (-3, 7, 2) and (2, 4, -1)
3. (-1, 3, -4) and (1, -3, 4)    