Given: X, Y, and Z are the midpoints of the sides of triangle ABC. Find rhe ratio of the area of triangle XYZ to the area of triangle ABC

To find the ratio of the area of triangle XYZ to the area of triangle ABC, we need to first understand a property of triangles involving midpoints.

The line segments connecting the midpoints of two sides of a triangle are called medians. A median divides the triangle into two smaller triangles of equal area.

In this case, X, Y, and Z are the midpoints of the sides of triangle ABC, which means triangle XYZ is formed by connecting the midpoints of triangle ABC. Therefore, triangle XYZ is a smaller triangle whose area is half the area of triangle ABC.

Hence, the ratio of the area of triangle XYZ to the area of triangle ABC is 1:2 (or 1/2).

To solve this problem algebraically, we can use the fact that the ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides.

Let us assume that the side lengths of triangle ABC are a, b, and c, and the side lengths of triangle XYZ are x, y, and z.

Since X, Y, and Z are the midpoints of the sides of triangle ABC, we have:
AX = BX = a/2
BY = CY = b/2
CZ = AZ = c/2

Now, we notice that triangle XYZ is similar to triangle ABC, with a similarity ratio of 1:2.

So, we can set up the following proportions:
x/a = 1/2
y/b = 1/2
z/c = 1/2

Now, to find the ratio of the areas, we square the ratios of the corresponding sides:

(x/a)^2 : (y/b)^2 : (z/c)^2
= (1/2)^2 : (1/2)^2 : (1/2)^2
= 1/4 : 1/4 : 1/4
= 1 : 1 : 1

Since the ratio of the areas is 1:1:1, we can conclude that the ratio of the area of triangle XYZ to the area of triangle ABC is 1:2 (or 1/2).