Question

the first figure in the sequence shown is an equilateral triangle. the next figure was formed from the first by connecting the midpoints 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 a square units
3 b square units
4 c 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 new smaller triangle (the center one) is $\frac{1}{4}$ of the area of the original triangle.

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

Since the area of the first triangle is $A_1 = 192$ square units, and the area of the center triangle in the second figure $A_2$ is related to the area of the first figure by $A_2=\frac{1}{4}A_1$. So $A_2=\frac{1}{4}\times192 = 48$ square units.

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

The center triangle in the third figure is formed from the center triangle of the second figure. Let the area of the center triangle in the second figure be $A_2 = 48$ square units. Then the area of the center triangle in the third figure $A_3=\frac{1}{4}A_2$. So $A_3=\frac{1}{4}\times48 = 12$ square units.

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

The center triangle in the fourth figure is formed from the center triangle of the third figure. Let the area of the center triangle in the third figure be $A_3 = 12$ square units. Then the area of the center triangle in the fourth figure $A_4=\frac{1}{4}A_3$. So $A_4=\frac{1}{4}\times12 = 3$ square units.

Answer:

A. 48 square units
B. 12 square units
C. 3 square units