Example Question - convex polygon
Here are examples of questions we've helped users solve.
Calculating Triangles in a Heptagon
The image features a convex polygon with seven sides, also known as a heptagon. To solve the problem of determining how many triangles are formed by drawing all the diagonals from a single vertex, you need to understand that a triangle is formed by connecting three non-collinear points. In a polygon, these points are the vertices. From a single vertex in a heptagon, you can draw diagonals to all other non-adjacent vertices. There are four vertices you can connect to with diagonals from one vertex (since the two adjacent vertices do not form diagonals but are the sides of the heptagon). Each diagonal you draw from the vertex will form a triangle with the segment of the side of the heptagon to each side of that diagonal. So for each non-adjacent vertex, you get one triangle when you connect it with a diagonal. Thus, for a seven-sided polygon (heptagon), drawing all the diagonals from one vertex to the four non-adjacent vertices will form four triangles.
Polygon Diagonals and Triangles
The image shows a convex polygon, and the question asks, "How many triangles are formed by drawing all the diagonals from a single vertex?" To solve this, you can count the number of sides (vertices) of the polygon and then use the following formula to find the number of triangles formed: Number of triangles formed from one vertex = Number of vertices (n) - 2 However, I cannot visually determine the exact number of sides of the polygon from the image provided. To count the number of sides (vertices), you should start at one corner and move around the polygon, counting each corner until you end up at the starting corner again. If you can provide the number of sides, I can help you calculate the number of triangles.
Number of Triangles in Polygon
The image shows a convex polygon. To determine how many triangles are formed by drawing all the diagonals from a single vertex, we follow a general rule: For any polygon with n sides (an n-gon), the number of diagonals from a single vertex to all non-adjacent vertices is (n - 3), because you cannot draw a diagonal to the vertex itself or to the two adjacent vertices. In the image provided, the convex polygon appears to be a pentagon, which has 5 sides. So we apply the rule: Number of triangles formed = n - 3 Number of triangles formed = 5 - 3 = 2 Therefore, by drawing all the diagonals from one vertex in the pentagon, we would form 2 triangles.
Calculating Sum of Interior Angles of Convex Polygon
To find the sum of the interior angles of a convex polygon, you can use the following formula: Sum of interior angles = (n - 2) × 180° where n is the number of sides of the polygon. The polygon in the image appears to be a triangle, which has 3 sides. So we substitute n = 3 into the formula: Sum of interior angles = (3 - 2) × 180° Sum of interior angles = 1 × 180° Sum of interior angles = 180° Therefore, the sum of the interior angle measures of this polygon is 180 degrees.
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