Draw every line of symmetry for the hexagon. For polygons with an even number of sides, all lines of symmetry pass through either - (pick two) two vertices \( \quad \) one vertex and one side two sides

For a regular hexagon, which has six sides, the lines of symmetry can be categorized based on the options provided. 1. **Two vertices**: A line of symmetry can be drawn connecting two opposite vertices. 2. **One vertex and one side**: A line of symmetry can also be drawn from a vertex to the midpoint of the opposite side. ### Step-by-step Explanation: 1. **Identify the vertices**: Label the vertices of the hexagon as \( A, B, C, D, E, F \). 2. **Draw lines through two vertices**: - Lines can be drawn from \( A \) to \( D \), \( B \) to \( E \), and \( C \) to \( F \). These lines will divide the hexagon into two mirror-image halves. 3. **Draw lines from one vertex to one side**: - A line can be drawn from vertex \( A \) to the midpoint of side \( BC \). - Similarly, lines can be drawn from \( B \) to the midpoint of side \( CD \), from \( C \) to the midpoint of side \( DE \), from \( D \) to the midpoint of side \( EF \), from \( E \) to the midpoint of side \( FA \), and from \( F \) to the midpoint of side \( AB \). ### Summary of Lines of Symmetry: - **Through two vertices**: \( AD, BE, CF \) - **Through one vertex and one side**: \( A \) to midpoint of \( BC \), \( B \) to midpoint of \( CD \), \( C \) to midpoint of \( DE \), \( D \) to midpoint of \( EF \), \( E \) to midpoint of \( FA \), \( F \) to midpoint of \( AB \) In total, a regular hexagon has 6 lines of symmetry. Now, I will draw the hexagon and its lines of symmetry. Simplify the expression by following steps: - step0: Solution: \(drawaregularhexagonwithlinesofsym\times metrythroughverticesandmidpointsofsides\) - step1: Rewrite the expression: \(d^{4}r^{6}a^{5}we^{7}g^{3}ulh^{4}xo^{5}n^{4}wi^{6}t^{5}ls^{6}fym^{3}yuvcpf\) - step2: Evaluate the power: \(d^{4}r^{6}a^{5}we^{7}g^{3}ulh^{4}xo^{5}n^{4}w\left(-1\right)t^{5}ls^{6}fym^{3}yuvcpf\) - step3: Use the rules for multiplication and division: \(-d^{4}r^{6}a^{5}we^{7}g^{3}ulh^{4}xo^{5}n^{4}wt^{5}ls^{6}fym^{3}yuvcpf\) - step4: Multiply the terms: \(-d^{4}r^{6}a^{5}w^{2}e^{7}g^{3}ulh^{4}xo^{5}n^{4}t^{5}ls^{6}fym^{3}yuvcpf\) - step5: Multiply the terms: \(-d^{4}r^{6}a^{5}w^{2}e^{7}g^{3}u^{2}lh^{4}xo^{5}n^{4}t^{5}ls^{6}fym^{3}yvcpf\) - step6: Multiply the terms: \(-d^{4}r^{6}a^{5}w^{2}e^{7}g^{3}u^{2}l^{2}h^{4}xo^{5}n^{4}t^{5}s^{6}fym^{3}yvcpf\) - step7: Multiply the terms: \(-d^{4}r^{6}a^{5}w^{2}e^{7}g^{3}u^{2}l^{2}h^{4}xo^{5}n^{4}t^{5}s^{6}f^{2}ym^{3}yvcp\) - step8: Multiply the terms: \(-d^{4}r^{6}a^{5}w^{2}e^{7}g^{3}u^{2}l^{2}h^{4}xo^{5}n^{4}t^{5}s^{6}f^{2}y^{2}m^{3}vcp\) - step9: Reorder the terms: \(-e^{7}d^{4}r^{6}a^{5}w^{2}g^{3}u^{2}l^{2}h^{4}xo^{5}n^{4}t^{5}s^{6}f^{2}y^{2}m^{3}vcp\) It seems that I cannot directly draw the hexagon and its lines of symmetry. However, I can describe how you can visualize or draw it yourself: 1. **Draw a regular hexagon**: Start by drawing a hexagon with vertices labeled \( A, B, C, D, E, F \) in a clockwise manner. 2. **Draw lines through two vertices**: - Draw a line from vertex \( A \) to vertex \( D \). - Draw a line from vertex \( B \) to vertex \( E \). - Draw a line from vertex \( C \) to vertex \( F \). 3. **Draw lines from one vertex to one side**: - Find the midpoint of side \( BC \) and draw a line from vertex \( A \) to this midpoint. - Repeat this for the other vertices: - From \( B \) to the midpoint of \( CD \) - From \( C \) to the midpoint of \( DE \) - From \( D \) to the midpoint of \( EF \) - From \( E \) to the midpoint of \( FA \) - From \( F \) to the midpoint of \( AB \) This will give you a total of 6 lines of symmetry for the hexagon. Each line divides the hexagon into two symmetrical halves.