Question

The vertices of a triangle A(-5,-1),B(3,-5),C(5,2). Show that the area of the is four times the area of the triangle formed by joining the mid-points of the sides of the triangle.

Answer 1

To show that the area of the triangle is four times the area of the triangle formed by the midpoints of the sides, we can use the formula for the area of a triangle and the properties of midpoints.

Step-by-step explanation:

To show that the area of the triangle is four times the area of the triangle formed by the midpoints of the sides, we can use the formula for the area of a triangle and the properties of midpoints.

Step 1: Find the coordinates of the midpoints of each side of the triangle. The midpoint of AB is (-1, -3), the midpoint of BC is (4, -1), and the midpoint of AC is (0, 0).

Step 2: Find the lengths of the sides of the triangles using the distance formula. The sides of ABC are AB = sqrt(58), BC = sqrt(74), and AC = sqrt(98). The sides of the triangle formed by the midpoints are AB' = sqrt(20), B'C' = sqrt(34), and AC' = sqrt(32).

Step 3: Use the formula for the area of a triangle (area = 1/2 * base * height) to calculate the area of both triangles. The area of ABC is 5.5 units^2 and the area of the triangle formed by the midpoints is 2.5 units^2.

Step 4: Calculate the ratio of the areas of the two triangles. 5.5 / 2.5 = 2.2. Therefore, the area of the triangle ABC is not four times the area of the triangle formed by the midpoints.