Free math worksheets

- Imagine folding each figure into two matching halves. - A symmetry fold must match every vertex and side. - Check more than just the top point.

1. Figure a) can be folded down the middle so its left and right sides match exactly. 2. Figure b) has unequal side positions, so no fold makes the two halves match. Therefore b) does not have a line of symmetry.

b)

- A valid fold must make every edge on one side land on an edge on the other side. - Look for the midpoint of the base. - Test the proposed lines by imagining an actual fold.

1. In a), the vertical line passes through the top vertex and the midpoint of the base, dividing the triangle into matching halves. 2. The horizontal line in b) does not match the top of the triangle with the base. 3. The slanted line in c) does not divide the triangle into mirror-image halves. Therefore a) is correct.

a)

- Try possible folds through the center of each figure. - A line counts only if every point has a matching point across it. - Different figures can have different numbers of valid folds.

1. Figure a) is a scalene triangle, so it has \(0\) lines of symmetry. 2. Figure b) is an isosceles triangle, so it has \(1\) line of symmetry through its top vertex and the midpoint of its base. 3. Figure c) is a non-square rectangle, so it has \(2\) lines of symmetry: one vertical and one horizontal through its center.

a) \(0\) b) \(1\) c) \(2\)

- Imagine folding each quadrilateral so that its edges match exactly. - Picture the quadrilaterals and test possible fold lines. - Distinguish lines through vertices from lines through side midpoints. - Which quadrilateral has the greatest number of matching fold lines?

1. A rhombus that is not a square has exactly two lines of symmetry: its diagonals. 2. A rectangle that is not a square has exactly two lines of symmetry. Each passes through the midpoints of a pair of opposite sides. 3. A square has four lines of symmetry: its two diagonals and the two lines through the midpoints of opposite sides.

a) A rhombus that is not a square. b) A rectangle that is not a square. c) A square. Its lines of symmetry are the two diagonals and the two lines through the midpoints of opposite sides.

- 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 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 fold lines.

1. A vertical fold through the top vertex and the midpoint of the bottom side maps the left half of the pentagon onto the right half. 2. No horizontal or diagonal fold maps the figure onto itself. 3. Therefore the pentagon has exactly one line of symmetry.

Yes. It has exactly one line of symmetry: the vertical line through the top vertex and the midpoint of the bottom side.

- 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

- Check where each vertex would land after a fold. - Compare the left and right vertices with the dashed line. - A valid symmetry line must match the entire boundary, not just one pair of points.

1. The dashed line passes through the top and bottom vertices. 2. The left vertex and right vertex are the same distance from the dashed line and lie at the same height. 3. Folding along the dashed line matches all four sides, so it is a line of symmetry.

Yes. Folding along the dashed vertical line maps the left half exactly onto the right half.

- Reflected points stay at the same height across a vertical fold line. - Count how far corresponding vertices are from the dashed line. - Every corresponding pair must match, not just the top and bottom points.

1. In a), each right-side vertex is directly across the fold line from a left-side vertex at the same height and the same distance, so the halves are reflections. 2. In b), the upper right vertex is shifted upward, so the halves do not match. 3. In c), the middle right vertex is too close to the fold line, so the distances do not match. Therefore a) is the exact reflection.

a)

- Across a horizontal fold, matching points have the same left-right position. - Compare each point's distance above or below the dashed line. - One mismatched vertex is enough to break symmetry.

1. Across a horizontal fold, reflected points stay directly above or below one another and keep the same distance from the fold line. 2. Drawing a) satisfies this for every vertex. 3. Drawing b) has one lower vertex too far from the fold, and drawing c) shifts one lower vertex sideways. Therefore a) is the exact reflection.

a)

- Reflections across a vertical line stay at the same height. - Count grid cells from each point to the dashed line. - Matching points must be equally far from the fold on opposite sides.

1. Points \(A\) and \(B\) are at the same height and each is \(3\) units from the fold line, on opposite sides. 2. Points \(C\) and \(D\) are at the same height, but their distances from the fold line are \(2\) units and \(1\) unit. 3. Points \(E\) and \(F\) are equally far from the fold line, but they are at different heights. Therefore \(A\) and \(B\) are reflections.

\(A\) and \(B\)

- Test vertical and horizontal folds first. - Then check whether either diagonal sends every corner to a matching corner. - Concave corners must match concave corners after a fold.

1. A vertical fold through the center matches the left and right halves. 2. A horizontal fold through the center matches the top and bottom halves. 3. A diagonal fold does not match the inward and outward corners. Therefore the figure has exactly \(2\) lines of symmetry.

\(2\)

- Imagine folds through opposite vertices. - Test both diagonals of each figure. - Count only folds that make the entire outline match.

1. The kite in a) has one line of symmetry: the vertical line through its top and bottom vertices. 2. The rhombus in b) has two lines of symmetry: its two diagonals. 3. Therefore the rhombus has one more line of symmetry than the kite.

a) \(1\) line b) \(2\) lines The rhombus has more.

- Keep the reflected point at the same height. - Count the grid cells from \(P\) to the dashed line. - Move the same number of cells to the other side.

1. Point \(P\) is \(2\) grid units left of the fold line. 2. Its reflection must be at the same height and \(2\) grid units to the right of the fold line. 3. Point \(R\) is in that position.

\(R\)

- Test vertical, horizontal, and diagonal folds for each figure. - The word exactly rules out figures with more than one valid fold. - Count only folds that make every edge match.

1. Figure a), an isosceles trapezoid, has exactly one vertical line of symmetry. 2. Figure b), a non-square rectangle, has two lines of symmetry. 3. Figure c), a scalene triangle, has no line of symmetry. 4. Figure d), a square, has four lines of symmetry. Therefore a) is the only figure with exactly one line of symmetry.

a)

- Imagine folding each kind of quadrilateral along possible mirror lines. - Compare the horizontal, vertical, and diagonal folds that work for a square and a non-square rectangle. - For a statement about every trapezoid, think about whether a lopsided trapezoid can match across any fold.

1. Statement a) is false. A rectangle that is not a square has exactly two lines of symmetry. A square has four. 2. Statement b) is true. A square has four lines of symmetry. 3. Statement c) is false. A trapezoid does not have to have a line of symmetry; a trapezoid with unequal nonparallel sides can have none.

a) False. A non-square rectangle has exactly two lines of symmetry; a square has four. b) True c) False. A trapezoid does not have to have a 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

- Connecting two vertices is not enough to make a symmetry line. - Think about which sides would land on each other after the fold. - Compare the side lengths that a diagonal fold would pair.

1. A line of symmetry must fold the entire rectangle onto itself. 2. Folding a non-square rectangle along a diagonal would try to match a long side with a short side. 3. Those sides have different lengths, so the halves do not match. Therefore a diagonal is not a line of symmetry for a non-square rectangle.

Maya is incorrect. A diagonal fold in a non-square rectangle would match a long side with a short side, so the two halves do not coincide.

- Check all three requirements, including the condition that diagonal symmetry must be absent. - A square and a non-square rectangle do not have the same symmetry count. - Test the kite's horizontal fold as well as its vertical fold.

1. Outline a), a non-square rectangle, has vertical and horizontal symmetry but no diagonal symmetry. 2. Outline b), a square, also has diagonal symmetry, so it does not meet the final condition. 3. Outline c), a kite, has only vertical symmetry. Therefore a) meets all three requirements.

a)

- A reflected point is the same perpendicular distance from the fold line on the opposite side. - A vertical fold changes left/right position but not height. - Track \(A\) and \(B\) separately before comparing their reflected positions.

1. Reflection keeps the perpendicular distance to the fold line, but moves the point to the opposite side. So \(A'\) is \(3\,\text{cm}\) to the right of the fold. 2. Likewise, \(B'\) is \(5\,\text{cm}\) to the right of the fold. 3. Reflection across a vertical line keeps vertical positions unchanged, so \(B'\) is still \(2\,\text{cm}\) higher than \(A'\).

\(A'\) is \(3\,\text{cm}\) to the right of the fold. \(B'\) is \(5\,\text{cm}\) to the right of the fold and is \(2\,\text{cm}\) higher than \(A'\).

- Check where both the inward corners and the outward points would land. - A fold must match every boundary feature, not just the center. - Test vertical, horizontal, and diagonal candidates independently.

1. In a), the vertical fold matches the left and right halves, so it is a line of symmetry. 2. In b), the horizontal fold matches the top and bottom halves, so it is a line of symmetry. 3. In c) and d), diagonal folds do not match the inward corners with inward corners and the outward points with outward points. Therefore only a) and b) are lines of symmetry.

a) and b)