The vertices of a triangle are A(-3, 5), B(7, 1), and C(3, 9). 1. Find the coordinates of the midpoint M of BC. 2. Show that AM is not a median of an equilateral triangle by finding the lengths of all three sides. 3. Find the area of $\triangle$ ABC. 4. A point P divides the line segment AB internally in the ratio 2:3. Find the coordinates of P. 5. Determine whether the point P lies inside, outside, or on the boundary of $\triangle$ ABC.

Step 1: Find the coordinates of the midpoint M of BC

To find the midpoint of a line segment, we use the midpoint formula. For two points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint is given by the average of their $x$ -coordinates and the average of their $y$ -coordinates. Here, we apply this formula to points B(7, 1) and C(3, 9).

Step 2: Calculate the lengths of the sides of $\triangle$ ABC

To determine if the triangle is equilateral, we need to find the lengths of all three sides using the distance formula. The distance formula for two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. We calculate the lengths of AB, BC, and AC.

Step 3: Determine if $\triangle$ ABC is equilateral

For a triangle to be equilateral, all three of its sides must have equal lengths. By comparing the calculated lengths of AB, BC, and AC, we can see that they are all different. Therefore, triangle ABC is not an equilateral triangle, and AM is not a median of an equilateral triangle.

Step 4: Find the area of $\triangle$ ABC

To find the area of a triangle given its vertices, we use the determinant formula. For vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the area is given by $\frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$. We substitute the coordinates of A, B, and C into this formula.

Step 5: Find the coordinates of point P dividing AB in ratio 2:3

To find the coordinates of point P that divides the line segment AB internally in the ratio $m:n$, we use the section formula. For points A $(x_1, y_1)$ and B $(x_2, y_2)$, and ratio $m:n$, the coordinates of P are given by $\left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right)$. Here, A is $(-3, 5)$, B is $(7, 1)$, $m=2$, and $n=3$.

Step 6: Determine if point P lies inside, outside, or on the boundary of $\triangle$ ABC

A point P lies on the boundary of a triangle if it lies on one of its sides. Since point P divides the line segment AB internally, it means P lies on the side AB of $\triangle ABC$. Therefore, P is on the boundary of the triangle. We can also verify this by checking if the sum of the areas of $\triangle PAC$ and $\triangle PBC$ equals the area of $\triangle ABC$. However, since P lies on AB, the area of $\triangle PAB$ is 0, and the point is on the boundary.

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 →