Question:

Find the lengths of the medians of the triangle with vertices A(0,0,6),B(0,4,0) and (6,0,0).

Answer:

Step 1: Calculate the midpoints of the sides of the triangle. The midpoint of AB can be calculated by taking the average of the x-coordinates and the average of the y-coordinates of the two points: Midpoint of AB: (0+6)/2, (0+4)/2, (6+0)/2 = (3,2,3) The midpoint of BC can be calculated by taking the average of the x-coordinates and the average of the y-coordinates of the two points: Midpoint of BC: (6+0)/2, (4+0)/2, (0+0)/2 = (3,2,0) The midpoint of AC can be calculated by taking the average of the x-coordinates and the average of the y-coordinates of the two points: Midpoint of AC: (0+6)/2, (4+0)/2, (6+0)/2 = (3,2,3)

Step 2: Calculate the distance between the midpoints and the vertices. The distance between the midpoint of AB and vertex A can be calculated using the distance formula: Distance between midpoint of AB and vertex A = √((3-0)^2 + (2-0)^2 + (3-6)^2) = 5 The distance between the midpoint of BC and vertex B can be calculated using the distance formula: Distance between midpoint of BC and vertex B = √((3-0)^2 + (2-4)^2 + (0-0)^2) = 2 The distance between the midpoint of AC and vertex C can be calculated using the distance formula: Distance between midpoint of AC and vertex C = √((3-6)^2 + (2-0)^2 + (3-0)^2) = 5

Step 3: Calculate the length of the medians. The length of the median from vertex A can be calculated by adding the distances between the midpoint of AB and vertex A, and the midpoint of AC and vertex A: Length of median from vertex A = 5 + 5 = 10 The length of the median from vertex B can be calculated by adding the distances between the midpoint of AB and vertex B, and the midpoint of BC and vertex B: Length of median from vertex B = 2 + 5 = 7 The length of the median from vertex C can be calculated by adding the distances between the midpoint of AC and vertex C, and the midpoint of BC and vertex C: Length of median from vertex C = 5 + 2 = 7

Therefore, the lengths of the medians of the triangle with vertices A(0,0,6), B(0,4,0) and C(6,0,0) are 10, 7 and 7 respectively.

Question:

A point R with x-coordinate 4 lies on the line segment joining the points P(2,−3,4) and Q(8,0,10). Find the coordinates of the point R.

Answer:

1. To find the coordinates of the point R we need to first find the coordinates of the midpoint of the line segment.

2. The midpoint of the line segment joining the points P and Q can be found by taking the average of the x, y, and z coordinates of P and Q.

3. The x-coordinate of the midpoint is (2+8)/2 = 5.

4. The y-coordinate of the midpoint is (-3+0)/2 = -1.5.

5. The z-coordinate of the midpoint is (4+10)/2 = 7.

6. Since we know that the point R lies on the line segment, and its x-coordinate is 4, then the coordinates of the point R are (4, -1.5, 7).

Question:

If the origin is the centroid of the triangle PQR with vertices P(2a,2,6),Q(−4,3b,−10) and R(8,14,2c), then find the values of a,b and c

Answer:

1. The centroid of a triangle is the point at the intersection of the three medians of the triangle.

2. The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

3. The midpoint of the side PQ is (−1, 2.5b, −8).

4. The midpoint of the side QR is (4, 8.5b, −4).

5. The midpoint of the side PR is (5a, 8, 4c).

6. Equate the x-coordinates of the three midpoints to find the value of a:

−1 = 5a a = −1/5

7. Equate the y-coordinates of the three midpoints to find the value of b:

2.5b = 8.5b b = 8/5

8. Equate the z-coordinates of the three midpoints to find the value of c:

−8 = 4c c = −2

Question:

If A and B be the points (3,4,5) and (−1,3,−7) respectively. Find the equation of the set of points P such that PA^2 +PB^2 =K^2, where K is a constant

Answer:

1. Let the coordinates of point P be (x, y, z).

2. We need to find the equation of the set of points P such that PA^2 +PB^2 =K^2.

3. We know that the coordinates of point A and B are (3, 4, 5) and (−1, 3, −7) respectively.

4. Therefore, the equation of the set of points P can be written as

(x - 3)2 + (y - 4)2 + (z - 5)2 + (-1 - x)2 + (3 - y)2 + (-7 - z)2 = K2

5. Simplifying the above equation, we get

2x2 + 2y2 + 2z2 - 12x - 8y + 12z - 28 = K2

6. Therefore, the equation of the set of points P such that PA^2 +PB^2 =K^2 is

2x2 + 2y2 + 2z2 - 12x - 8y + 12z - 28 = K2

Question:

Three vertices of a parallelogram ABCD are A(3,−1,2),B(1,2,−4) and C(−1,1,2). Find the coordinates of the fourth vertex.

Answer:

1. The fourth vertex of the parallelogram is D.
2. Since the parallelogram is a closed figure, the sum of the x-coordinates of the four vertices must be equal. Therefore, 3 + 1 + (-1) + Dx = 0
3. Similarly, the sum of the y-coordinates must also be equal. Therefore, -1 + 2 + 1 + Dy = 0
4. Lastly, the sum of the z-coordinates must be equal. Therefore, 2 + (-4) + 2 + Dz = 0
5. Solving the above three equations, we get Dx = -3, Dy = -3 and Dz = 6
6. Therefore, the coordinates of the fourth vertex D are (-3,-3,6).

Question:

Find the coordinates of a point on y-axis which are at a distance of 5√2 from the point P(3,−2,5)

Answer:

Step 1: The coordinates of the point P are given as (3, -2, 5).

Step 2: Since the point is on the y-axis, the x-coordinate of the point will be 0.

Step 3: The distance between the points P and the point on the y-axis is 5√2.

Step 4: We can use the Pythagorean Theorem to calculate the y-coordinate of the point.

Step 5: The equation for the Pythagorean Theorem is a2 + b2 = c2.

Step 6: Substituting the values, we get (-2)2 + y2 = (5√2)2

Step 7: Simplifying the equation, we get y2 = 82

Step 8: Taking the square root of both sides, we get y = ±√82

Step 9: The coordinates of the point on the y-axis which are at a distance of 5√2 from the point P(3,−2,5) are (0, ±√82).