Free Math Worksheets

- Imagine folding each shape so its edges match. - Compare folds through side midpoints with folds through opposite vertices. - Consider what changes when all four sides are congruent.

1. A non-square rectangle has \(2\) lines of symmetry. Each passes through the midpoints of a pair of opposite sides. 2. A square has \(4\) lines of symmetry. 3. A square has the same two midpoint lines as a rectangle, and its two diagonals are also lines of symmetry because all four sides are congruent.

a) \(2\); they pass through the midpoints of opposite sides. b) \(4\) c) The square’s diagonals are also lines of symmetry.

- Imagine folding the figure along a line so that its edges match exactly. - Look for pairs of vertices that are the same distance from a possible fold line. - Test vertical, horizontal, and diagonal lines.

1. Reflecting the figure across the vertical line \(x = 2\) maps the left half onto the right half. 2. This line passes through the top vertex \((2, 4)\) and the midpoint of the bottom side \((2, 0)\). 3. No horizontal or diagonal line maps the figure onto itself, so there is exactly one line of symmetry.

Yes. The figure has exactly one line of symmetry: the vertical line \(x = 2\).

- Imagine folding along the red line. - Matching points must be the same perpendicular distance from the fold line. - Check that the matching half has the reversed orientation. - A vertical shift prevents the halves from matching.

1. For the red line to be a line of symmetry, every point in the matching half must be the same perpendicular distance from the red line as its partner in figure O. 2. Image 1 has the correct overall position, but it keeps the original orientation instead of reflecting it across the red line. 3. Image 2 reverses the orientation and places every corresponding point the same distance from the red line on the opposite side. 4. Image 3 is shifted 1 unit upward, so the two halves would not match when folded along the red line.

Image 2

- Find the tallest stack in each plan column. - Compare the left and right heights in the front view.

1. Find the greatest stack height in each column of the plan. The front-view heights are \(\max(1,2,3)=3\), \(\max(4,0,1)=4\), and \(\max(1,2,3)=3\). 2. The left and right columns have the same height, so the front view matches across a vertical line through the middle column.

Yes, the front view has a vertical line of symmetry.

- Imagine folding each figure so that its edges match exactly. - A line of symmetry divides a figure into mirror-image halves. - Check lines through vertices and lines through side midpoints. - For regular polygons, look for repeated symmetry around the center.

1. Figure a) is a rectangle. It has \(2\) lines of symmetry: one horizontal and one vertical. 2. Figure b) is an equilateral triangle. It has \(3\) lines of symmetry, each passing through a vertex and the midpoint of the opposite side. 3. Figure c) is an isosceles trapezoid. It has \(1\) vertical line of symmetry. 4. Figure d) is a regular hexagon. It has \(6\) lines of symmetry: \(3\) through pairs of opposite vertices and \(3\) through midpoints of opposite sides.

a) \(2\) lines of symmetry b) \(3\) lines of symmetry c) \(1\) line of symmetry d) \(6\) lines of symmetry

- First find the three heights in the front view. - Decide what must change so the left and right heights match. - Each plan must add exactly one cube. - Check the front view made by each plan.

1. The front-view heights of building G are \(\max(1,2)=2\), \(\max(2,3)=3\), and \(\max(1,1)=1\), so the front view is \((2, 3, 1)\). 2. To make the view symmetric, the right visible column must increase to height \(2\), giving \((2, 3, 2)\). 3. Plan A raises the front-right stack from \(1\) to \(2\), so its front view is \((2, 3, 2)\). Plan B leaves the front view unchanged, Plan C gives \((3, 3, 1)\), and Plan D gives \((2, 4, 1)\).

Plan A