Question

the first figure in the sequence shown is an equilateral triangle. the next figure was formed from the first by connecting the mid - points of each side of the first triangle to form four new triangles—one triangle in the center that points downward, and three triangles that point upward. this process was repeated to form the second and third figures. if the area of the first figure is 192 square units, complete the table to show the areas of the center triangle in each of the next three figures.
figure center triangle area
1 192 square units
2 square units
3 square units
4 square units

Explanation:

Step1: Understand the area - ratio relationship

When we connect the mid - points of the sides of a triangle to form four new triangles, the area of the central triangle is $\frac{1}{4}$ of the area of the original triangle.

Step2: Calculate the area of the center triangle in Figure 2

The area of the first figure is $A_1 = 192$ square units. For Figure 2, the area of the center triangle $A_2=\frac{1}{4}\times A_1$. Substituting $A_1 = 192$ into the formula, we get $A_2=\frac{1}{4}\times192 = 48$ square units.

Step3: Calculate the area of the center triangle in Figure 3

The center triangle of Figure 3 is formed from the center triangle of Figure 2 in the same way. So $A_3=\frac{1}{4}\times A_2$. Since $A_2 = 48$ square units, then $A_3=\frac{1}{4}\times48 = 12$ square units.

Step4: Calculate the area of the center triangle in Figure 4

The center triangle of Figure 4 is formed from the center triangle of Figure 3 in the same way. So $A_4=\frac{1}{4}\times A_3$. Since $A_3 = 12$ square units, then $A_4=\frac{1}{4}\times12 = 3$ square units.

Answer:

A. 48
B. 12
C. 3