make solution for this problem---**Problem Heading:** Problem **Question Stem:** The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire 4 × 4 grid is covered by the star? **Chart/Diagram Description:** * **Type:** Geometric pattern within a grid. * **Main Elements:** * **Grid:** A square grid composed of 4 rows and 4 columns of identical smaller squares, totaling 16 small squares. * **Eight-pointed Star:** A star pattern is formed by shaded regions within this 4 × 4 grid. * **Shading:** * **Dark Gray Area:** The central 2x2 block of the grid (composed of 4 small squares) is entirely covered by a dark gray shaded region. This forms the central body of the star. * **Light Gray Area:** Extending outwards from the central dark gray block are eight triangular shapes, shaded in a lighter gray. Each of these triangles occupies exactly half of one small grid square. * Specifically, there are two light gray triangles in the top row (occupying the bottom half of the second and third squares from the left). * Two light gray triangles in the bottom row (occupying the top half of the second and third squares from the left). * Two light gray triangles in the leftmost column (occupying the right half of the second and third squares from the top). * Two light gray triangles in the rightmost column (occupying the left half of the second and third squares from the top). * **Unshaded Area:** The four corner squares of the 4x4 grid are entirely unshaded (white). Additionally, the remaining halves of the 8 squares containing the light gray star points are also unshaded (white). * **Lines:** The pattern is defined by straight lines forming the grid boundaries and diagonal lines within individual squares to delineate the star's shape. **Options:** (A) 40 (B) 50 (C) 60 (D) 75 (E) 80

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Let's analyze this eight-pointed star quilting pattern. We have a 4 by 4 grid containing 16 equal squares, numbered 1 through 16. The star consists of a dark gray central region covering the middle 2 by 2 area, plus eight light gray triangular points extending outward. Each triangular point occupies exactly half of a unit square. Our goal is to find what percentage of the total grid area is covered by this star pattern. Let's systematically analyze the grid structure. We have a 4 by 4 grid containing exactly 16 unit squares. Each square represents 1 unit of area, so the total grid area is 16 units. The grid is organized into 4 rows and 4 columns. To find the percentage covered by the star, we need to calculate the total area covered by both the dark gray central region and the light gray triangular points, then express this as a fraction of the total 16 unit squares. Now let's focus on the central star body, which is the dark gray region. This central area forms a 2 by 2 block consisting of exactly 4 complete unit squares. These are squares number 6, 7, 10, and 11 in our grid. Since each square has an area of 1 unit, the central star body has a total area of 4 unit squares. As a percentage of the total grid, this represents 4 out of 16 squares, which equals one-fourth or 25 percent of the entire grid area. Now let's analyze the eight triangular star points that extend from the central body. These light gray triangles are systematically distributed: 2 triangles in the top row, 2 in the bottom row, 2 in the left column, and 2 in the right column. Each triangle occupies exactly half of a unit square. Since we have 8 triangles, and each has an area of one-half unit square, the total area of all star points is 8 times one-half, which equals 4 unit squares. Now let's calculate the total area covered by the star. We combine the central body area and the star points area. The central body contributes 4 unit squares, and the 8 triangular star points contribute 8 times one-half, which equals 4 unit squares. Therefore, the total star area is 4 plus 4, which equals 8 unit squares. To find the percentage, we divide 8 by the total grid area of 16, then multiply by 100 percent. This gives us 8 divided by 16 times 100 percent, which equals 50 percent. The answer is option B: 50.