This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the geometry problems. Question 6. Define the term polygon. Give two examples of polygons. Definition: A polygon is a closed two-dimensional shape made up of straight line segments connected end-to-end to form a boundary. The line segments are called sides, and the points where they meet are called vertices*. Examples: Triangle Square Question 7. Find the measure of each interior angle of a regular hexagon. Step 1: Determine the number of sides for a hexagon. A hexagon has n=6 sides. Step 2: Calculate the sum of the interior angles of a polygon using the formula (n-2) × 180^. Sum of interior angles = (6-2) × 180^ Sum of interior angles = 4 × 180^ Sum of interior angles = 720^ Step 3: For a regular polygon, all interior angles are equal. Divide the sum of interior angles by the number of sides to find the measure of each interior angle. Each interior angle = Sum of interior anglesn Each interior angle = (720^)/(6) Each interior angle = 120^ The measure of each interior angle of a regular hexagon is 120^. Question 8. A polygon has an interior angle sum of 1260^. Calculate the number of sides it has. Step 1: Use the formula for the sum of interior angles of a polygon, which is (n-2) × 180^, where n is the number of sides. (n-2) × 180^ = 1260^ Step 2: Divide both sides by 180^. n-2 = (1260)/(180) n-2 = 7 Step 3: Add 2 to both sides to solve for n. n = 7 + 2 n = 9 The polygon has 9 sides. Question 9. One angle of a regular polygon measures 150^. Find the number of sides. Step 1: For a regular polygon, each interior angle and its corresponding exterior angle sum to 180^. Calculate the measure of one exterior angle. Exterior Angle = 180^ - Interior Angle Exterior Angle = 180^ - 150^ Exterior Angle = 30^ Step 2: The sum of the exterior angles of any polygon is 360^. For a regular polygon, all exterior angles are equal. The number of sides n can be found by dividing 360^ by the measure of one exterior angle. n = (360^)/(Exterior Angle) n = (360^)/(30^) n = 12 The regular polygon has 12 sides. Question 10. Draw a regular Octagon. (As an AI, I cannot physically draw. Below are the steps to construct a regular octagon using a compass and ruler.) To draw a regular octagon: 1. Draw a Circle: Use a compass to draw a circle. Mark its center point. 2. Draw Perpendicular Diameters: Draw a horizontal diameter and a vertical diameter through the center of the circle. These two diameters will divide the circle into four 90^ sectors. 3. Bisect the Angles: Bisect each of the four 90^ angles. To do this, you can place the compass point on the center, open it to any convenient radius, and draw arcs that intersect the circle and the existing diameters. Then, from the intersection points on the circle, draw two more arcs that intersect each other within each 90^ sector. Draw a line from the center through this intersection point to the circumference. This will create lines that divide each 90^ angle into two 45^ angles. 4. Mark Vertices: You will now have 8 equally spaced points on the circumference of the circle where the lines intersect the circle. These points are the vertices of the regular octagon. 5. Connect the Vertices: Use a ruler to connect these 8 points consecutively with straight line segments. These segments will form the 8 equal sides of the regular octagon.