The vertices of a triangle are $A(5, -1)$, $B(-3, -2)$, and $C(1, 8)$. Find the length of the median drawn from vertex $A$ to side $BC$, and determine the coordinates of the centroid of $\triangle ABC$.

Step 1: Find the midpoint of BC

To find the median from vertex A to side BC, we first need to find the midpoint of side BC. Let's call this midpoint D. We use the midpoint formula, which averages the x-coordinates and y-coordinates of points B and C.

Step 2: Calculate coordinates of D

Substitute the coordinates of B $(-3, -2)$ and C $(1, 8)$ into the midpoint formula. This gives us the coordinates of point D, which is $(-1, 3)$.

Step 3: Calculate the length of the median AD

Now that we have the coordinates of A $(5, -1)$ and D $(-1, 3)$, we can find the length of the median AD using the distance formula. The distance formula calculates the length of a line segment between two points.

Step 4: Calculate length of AD

Substitute the coordinates of A $(5, -1)$ and D $(-1, 3)$ into the distance formula. After performing the calculations, we find the length of the median AD to be $\sqrt{52}$, which simplifies to $2\sqrt{13}$ units.

Step 5: Find the coordinates of the centroid

The centroid of a triangle is the point where the three medians intersect. Its coordinates are found by averaging the x-coordinates and y-coordinates of all three vertices of the triangle.

Step 6: Calculate coordinates of the centroid G

Substitute the coordinates of A $(5, -1)$, B $(-3, -2)$, and C $(1, 8)$ into the centroid formula. After summing the coordinates and dividing by 3, we find the centroid G to be $(1, \frac{5}{3})$.

More from this chapter

• Q7. If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b. [2 Marks]
• Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6).
• Find a point on the y-axis that is at the same distance from the points (-5, 2) and (9, -2).
• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
• 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)
• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
• Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

All Coordinate Geometry solutions →