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Classify shapes by attributes

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5404693
A mystery shape is closed and has exactly \(5\) straight sides. What is the shape called?

Hints

- Count the straight sides in the clue. - Match that number of sides to a shape name.

Solution

1. A closed shape with \(5\) straight sides is a pentagon.

Answer

The shape is a pentagon.
5404743
Which shape is not a polygon: a triangle, a circle, a pentagon, or a hexagon? Explain why.

Hints

- Recall what kind of sides a polygon must have. - Check each choice for any curved edge.

Solution

1. Polygons are closed shapes made only of straight sides. 2. A circle has a curved edge, so it is not a polygon. 3. The triangle, pentagon, and hexagon have only straight sides.

Answer

The circle is not a polygon because it has a curved edge instead of straight sides.
5404793
Sort these shapes by number of sides: triangle, hexagon, and quadrilateral. Write them from fewest sides to most sides.

Hints

- Recall how many sides each shape name represents. - Put the three side counts in increasing order.

Solution

1. A triangle has \(3\) sides. 2. A quadrilateral has \(4\) sides. 3. A hexagon has \(6\) sides. 4. The order from fewest to most is triangle, quadrilateral, hexagon.

Answer

Triangle, quadrilateral, hexagon.
5405063
Sort the shapes into two groups: triangle, square, circle, and hexagon. Group 1 has only straight sides. Group 2 has a curved boundary.

Hints

- Check the boundary of each shape. - Put every shape with no curved part in the same group. - A circle is not a polygon because its boundary is curved.

Solution

1. A triangle, square, and hexagon are made only of straight sides. 2. A circle has a curved boundary. 3. Group 1 contains triangle, square, and hexagon. Group 2 contains circle.

Answer

Group 1: triangle, square, hexagon Group 2: circle
5405293
What familiar shape is shown? Explain which attributes you used.
Figure for problem 540529

Hints

- Look for any straight part of the boundary. - Check whether the figure has any corners. - Name the familiar shape that is perfectly round.

Solution

1. The figure has one curved boundary and no vertices. 2. It is perfectly round. 3. These attributes describe a circle.

Answer

A circle.
5508553
Which descriptions must be polygons? Choose all that apply. A. A closed shape with \(4\) straight sides B. A closed shape with \(5\) straight sides C. A circle D. An open path made of \(3\) straight segments

Hints

- Check whether each description is closed. - Then check whether every boundary piece is straight.

Solution

1. A polygon must be closed and made only of straight sides. 2. A and B meet both conditions. 3. C has a curved boundary, and D is open.

Answer

A and B.
5404853
Kai makes a closed shape with \(4\) boundary pieces, but one piece is curved. Kai calls it a quadrilateral. Is that name correct? Explain.

Hints

- Check more than the number of boundary pieces. - Recall what kind of sides every polygon must have. - Decide whether a curved piece can be one of a quadrilateral’s sides.

Solution

1. A quadrilateral is a closed shape with \(4\) straight sides. 2. Kai’s shape has a curved boundary piece instead of four straight sides. 3. Therefore, the shape is not a quadrilateral.

Answer

No. A quadrilateral must have \(4\) straight sides, and Kai’s shape has a curved boundary piece.
5404963
Which figure, a) or b), is a pentagon? Explain.
Figure for problem 540496

Hints

- Count the straight segments in each figure. - Check whether the first and last segments meet. - A polygon must be both closed and made of straight sides.

Solution

1. A pentagon must be closed and have \(5\) straight sides. 2. Figure a) is open because its first and last segments do not meet. 3. Figure b) is closed and has \(5\) straight sides, so it is a pentagon.

Answer

Figure b) is the pentagon.
5405103
A card says, “Hexagon: a polygon with \(5\) sides.” What is wrong, and how should the card be corrected?

Hints

- Compare the shape name with the stated side count. - Recall the names for polygons with \(5\) and \(6\) sides.

Solution

1. A polygon with \(5\) sides is a pentagon. 2. A hexagon has \(6\) sides. 3. The card should say, “Hexagon: a polygon with \(6\) sides.”

Answer

The side count is wrong. A hexagon has \(6\) sides; a \(5\)-sided polygon is a pentagon.
5405123
Some people call b) a “diamond.” Is b) a square? Explain using its sides and corners.
Figure for problem 540512

Hints

- Compare the side lengths before and after the turn. - Check whether the corners changed when the shape was rotated. - A shape’s orientation does not change its geometric attributes.

Solution

1. Turning a shape does not change its side lengths or corner sizes. 2. Figure b) still has \(4\) equal sides and \(4\) square corners. 3. Therefore, figure b) is still a square. “Diamond” describes how it looks, not a different geometric category.

Answer

Yes. Figure b) is still a square because it has \(4\) equal sides and \(4\) square corners.
5405163
Compare a) and b). Do they belong to the same shape category? Explain using their sides and corners.
Figure for problem 540516

Hints

- Compare the number and relative lengths of the sides. - Check the corners of both shapes. - Size is not one of the defining attributes of a square.

Solution

1. Both shapes have \(4\) equal sides and \(4\) square corners. 2. Changing the size does not change those defining attributes. 3. Both shapes are squares.

Answer

Yes. Both are squares because size does not change their defining side and corner attributes.
5405273
A closed polygon has \(6\) straight sides. Some sides are long and some are short. Does it still have to be a hexagon? Explain.

Hints

- Focus on the attribute used in the definition of a hexagon. - Decide whether equal side lengths are required by that definition.

Solution

1. A hexagon is defined as a polygon with \(6\) sides. 2. The side lengths do not all have to be equal. 3. The shape is still a hexagon.

Answer

Yes. Any closed polygon with \(6\) straight sides is a hexagon, even when the side lengths differ.
5405313
Shape A is a closed polygon with \(4\) equal straight sides. Shape B is a closed polygon with \(4\) straight sides of different lengths. What broad category contains both shapes?

Hints

- Ignore the difference in side lengths at first. - Find the side-count attribute the shapes share. - Name the broad category defined by that count.

Solution

1. Shape A has \(4\) sides. 2. Shape B also has \(4\) sides. 3. Every closed polygon with \(4\) straight sides is a quadrilateral. 4. Both shapes belong to the quadrilateral category.

Answer

Both shapes are quadrilaterals.
5405393
Maya has drawn one connected path using exactly \(3\) straight segments joined end to end. Which extra condition guarantees that the path forms a triangle? A. The last endpoint joins the first endpoint, enclosing one region. B. All \(3\) segments have the same length. C. One segment is horizontal. D. At least one corner is square.

Hints

- The drawing already has the exact number of straight segments needed for a triangle. - Ask what still has to happen for those segments to make a polygon rather than an open path. - Side length, direction, and one corner type do not determine whether the path is closed.

Solution

1. The path already has exactly \(3\) straight segments. 2. To form a triangle, those segments must make a closed polygon. 3. Choice A closes the path by joining the last endpoint to the first and makes one enclosed region. 4. Equal lengths, a horizontal segment, or a square corner do not by themselves close an open path.

Answer

A. The last endpoint joins the first endpoint, enclosing one region.
5405413
Is the figure a pentagon? Explain.
Figure for problem 540541

Hints

- Trace the boundary and count each straight side once. - Check whether the shape is closed. - A polygon does not need every vertex to point outward.

Solution

1. A pentagon is a closed polygon with \(5\) straight sides. 2. The figure is closed and still has \(5\) straight sides. 3. A vertex pointing inward does not change the side count, so the figure is still a pentagon.

Answer

Yes. It is still a pentagon because it is closed and has \(5\) straight sides.
5405423
Which description must be true for every quadrilateral? A. It is closed and has \(4\) straight sides. B. It has \(4\) equal sides. C. It has \(4\) square corners. D. It is wider than it is tall.

Hints

- Look for the attributes shared by every shape in the category. - Do not choose an attribute that belongs only to a special kind of quadrilateral. - Ignore how the shape is turned or stretched.

Solution

1. The defining attributes of a quadrilateral are that it is closed and has \(4\) straight sides. 2. Equal sides and square corners belong only to some quadrilaterals. 3. Width and height do not define the category. 4. Therefore, choice A must be true.

Answer

A. It is closed and has \(4\) straight sides.
5405473
A shape bot follows one instruction list. Which list guarantees a hexagon? A. Make a closed polygon with exactly \(6\) straight sides. B. Place \(6\) vertices, connect only \(5\) pairs, and leave the shape open. C. Make a closed polygon with exactly \(5\) straight sides.

Hints

- Recall the exact side count for a hexagon. - Check whether each choice guarantees a closed polygon. - A guarantee must rule out shapes with the wrong number of sides.

Solution

1. A hexagon is a closed polygon with exactly \(6\) straight sides. 2. Choice A states those defining attributes directly. 3. Choice B is open, and choice C has only \(5\) sides. 4. Therefore, choice A guarantees a hexagon.

Answer

A. Make a closed polygon with exactly \(6\) straight sides.
5405593
Shape A is a closed polygon with \(3\) straight sides. Shape B is a closed polygon with \(4\) straight sides. Each shape has perimeter \(12\,\text{units}\). a) Name the polygon category of Shape A and Shape B. b) Does matching perimeter put the two shapes in the same polygon category? Explain which attribute decides the category.

Hints

- Use the number of straight sides to name each polygon. - Compare the attribute used in the category names with the perimeter information. - For part b), explain why one matching measurement does not change the side counts.

Solution

1. Shape A has \(3\) straight sides, so it is a triangle. 2. Shape B has \(4\) straight sides, so it is a quadrilateral. 3. These polygon categories are determined by the number of sides, not by perimeter. 4. The equal perimeters therefore do not put the two shapes in the same category.

Answer

a) Shape A is a triangle. Shape B is a quadrilateral. b) No. Their side counts are different, and side count determines these polygon categories; matching perimeter does not.
5405633
Use the figure. a) How many sides and vertices does the shape have? b) Based only on its number of sides, which category fits the shape: triangle, quadrilateral, pentagon, or hexagon?
Figure for problem 540563

Hints

- Trace the outside boundary once. - Count each straight side and each place where two sides meet. - For part b), use only the side count, not a more specific quadrilateral name.

Solution

1. Trace the boundary once. It has \(4\) straight sides. 2. The \(4\) sides meet at \(4\) vertices. 3. A polygon with \(4\) sides is in the quadrilateral side-count category.

Answer

a) \(4\) sides and \(4\) vertices b) Quadrilateral
5508563
Which figures are quadrilaterals? Choose all that apply.
Figure for problem 550856

Hints

- Trace each boundary once. - Count only the straight sides on the outside boundary. - Select every figure with exactly four sides.

Solution

1. A quadrilateral is a closed polygon with \(4\) straight sides. 2. Figure a) has \(3\) sides, b) has \(4\), c) has \(5\), and d) has \(4\). 3. Therefore, b) and d) are quadrilaterals.

Answer

Figures b) and d).
5508573
Figures a) and b) look different. What broad shape category contains both figures? Explain using a shared attribute.
Figure for problem 550857

Hints

- Ignore the different slants and side lengths at first. - Count the outside sides of each figure. - Name the larger category defined by that shared count.

Solution

1. Figure a) has \(4\) straight sides. 2. Figure b) also has \(4\) straight sides. 3. Every closed polygon with \(4\) straight sides is a quadrilateral.

Answer

Both figures are quadrilaterals because each has \(4\) straight sides.
5508583
A teacher wants one rule that includes triangles, quadrilaterals, pentagons, and hexagons but does not include circles. Which rule works best? A. Shapes with at least \(3\) corners B. Closed shapes made only of straight sides C. Shapes with equal side lengths

Hints

- Look for a rule that is true for every listed polygon category. - Make sure the rule also excludes a circle. - Do not require an attribute that only some polygons have.

Solution

1. The listed polygon categories are all closed and made only of straight sides. 2. A circle does not satisfy that rule because its boundary is curved. 3. Rule A could describe a closed shape with curved boundary pieces, and C is not true for every listed polygon.

Answer

B. Closed shapes made only of straight sides.
5404923
Nora wants a closed polygon made of straight sides with only \(2\) vertices. Can such a polygon exist? Explain.

Hints

- Think of the polygon with the fewest possible sides. - Compare its number of vertices with the requested number.

Solution

1. A polygon needs at least \(3\) straight sides to enclose a region. 2. A polygon with \(3\) sides also has \(3\) vertices. 3. A closed polygon with only \(2\) vertices cannot exist.

Answer

No. A polygon needs at least \(3\) sides and \(3\) vertices.
5404983
A student describes a polygon as having \(6\) sides but only \(5\) vertices. Is that description possible? Explain.

Hints

- Think about what happens where two neighboring sides meet. - Compare the side count and vertex count of familiar polygons.

Solution

1. In a polygon, each pair of neighboring sides meets at a vertex. 2. Therefore, a polygon has the same number of sides and vertices. 3. A \(6\)-sided polygon must have \(6\) vertices, so the description is not possible.

Answer

No. A polygon with \(6\) sides must also have \(6\) vertices.
5405033
A closed polygon has exactly \(4\) vertices. What broad shape category must it belong to? Can you name a more specific category from this information alone?

Hints

- Connect the number of vertices to the number of sides. - Decide what the side count guarantees. - Notice which attributes would be needed for a more specific name.

Solution

1. A polygon with \(4\) vertices also has \(4\) sides. 2. Every \(4\)-sided polygon is a quadrilateral. 3. No side-length or corner information is given, so a more specific category cannot be determined.

Answer

It must be a quadrilateral. A more specific category cannot be determined from the information given.
5405213
Which statements are true for every polygon? Choose all that apply. A. It is closed. B. It has only straight sides. C. It has the same number of sides and vertices. D. It has exactly \(4\) sides.

Hints

- Think about the features shared by triangles, quadrilaterals, and pentagons. - Reject any statement that describes only one polygon category.

Solution

1. Every polygon is closed, so A is true. 2. Every polygon is made of straight sides, so B is true. 3. Each pair of neighboring sides meets at one vertex, so C is true. 4. Polygons can have different numbers of sides, so D is false.

Answer

A, B, and C.
5405453
An octagon has \(8\) sides. One corner is clipped off with one straight cut while the rest of the boundary stays in place. Without drawing the new shape, how many sides will it have? Explain what changes along the boundary.

Hints

- Focus only on the part of the boundary near the corner that is removed. - Decide whether the two sides beside that corner disappear or merely become shorter. - Account for the new straight cut as part of the outside boundary.

Solution

1. The octagon starts with \(8\) sides. 2. Clipping off one corner shortens the two sides beside that corner but does not remove them. 3. The straight cut creates one new boundary side. 4. The side count increases by \(1\), from \(8\) to \(9\).

Answer

\(9\) sides. The two old sides stay as shorter sides, and the cut adds one new side.
5405483
A hexagon has \(6\) sides. One vertex is removed by replacing the two sides that meet there with one straight side connecting the neighboring vertices. Without drawing the new shape, how many sides will it have, and what polygon will it be? Explain.

Hints

- Compare how many boundary sides are removed with how many replace them. - Work out the change in side count before naming the new polygon. - Match the resulting number of straight sides to a polygon name.

Solution

1. The hexagon starts with \(6\) sides. 2. Two boundary sides are replaced by one boundary side. 3. The side count decreases by \(1\), from \(6\) to \(5\). 4. A closed polygon with \(5\) straight sides is a pentagon.

Answer

\(5\) sides; the new shape is a pentagon.
5405493
A closed shape has \(10\) straight sides. Which statements are true? A. It is a polygon. B. It is a hexagon. C. It is a quadrilateral. D. It has \(10\) vertices.

Hints

- Use the closed and straight-side clues to identify the broad category. - Compare \(10\) with the side counts named in the choices. - Relate the number of polygon sides to its number of vertices.

Solution

1. A closed shape made only of straight sides is a polygon, so A is true. 2. A hexagon has \(6\) sides, and a quadrilateral has \(4\) sides, so B and C are false. 3. A polygon has the same number of vertices as sides, so the shape has \(10\) vertices and D is true.

Answer

A and D.
5405553
What is the name of the whole polygon? Does the inside segment change that name? Explain.
Figure for problem 540555

Hints

- Trace only the outside boundary first. - Decide whether the inside segment is part of that boundary. - Name the polygon from the number of outside sides.

Solution

1. A polygon category is determined by the sides on its outside boundary. 2. The outside boundary still has \(5\) straight sides. 3. The inside segment does not become a new side of the whole shape. 4. Therefore, the whole shape is still a pentagon.

Answer

No. The inside segment does not change the name. The outside boundary has \(5\) straight sides, so the whole shape is a pentagon.
5405573
Milo traces a closed polygon with \(12\) sides. He says it has only \(11\) vertices because the first and last side meet at the starting point. Is Milo correct? Explain.

Hints

- Think about how vertices are counted around a closed boundary. - Do not count the starting point twice, but do count it once. - Compare the final vertex count with the side count.

Solution

1. In a closed polygon, each side ends at a vertex and the next side begins there. 2. The starting point is one vertex, not an extra repeated vertex. 3. A polygon has the same number of sides and vertices. 4. Therefore, a \(12\)-sided polygon has \(12\) vertices, so Milo is not correct.

Answer

No. The polygon has \(12\) vertices.
5405623
A shape is closed and has \(5\) corners where boundary pieces meet. The description does not say whether every boundary piece is straight. Can you be sure the shape is a pentagon? Explain.

Hints

- List every defining attribute of a pentagon. - Compare that list with the information actually given. - Look for a missing condition about the boundary pieces.

Solution

1. A pentagon is a closed polygon with \(5\) straight sides. 2. Five meeting corners do not prove that every boundary piece is straight. 3. If even one boundary piece is curved, the shape is not a polygon and cannot be a pentagon. 4. Therefore, the information is not enough to be sure the shape is a pentagon.

Answer

No. You also need to know that all \(5\) boundary pieces are straight sides.
5508593
A mystery shape is a closed polygon with \(4\) straight sides. One pair of sides has equal length. Can you tell whether the shape is a rectangle, a rhombus, or a square? Explain what you can know for sure.

Hints

- Separate the attribute that is guaranteed from the attributes that are missing. - Ask what you would need to know about all four sides. - Ask what you would need to know about the corners.

Solution

1. Four straight sides guarantee that the shape is a quadrilateral. 2. One equal pair does not tell whether all four sides are equal. 3. No information is given about whether all four corners are square. 4. Therefore, no more specific category is guaranteed.

Answer

You can know only that it is a quadrilateral. The information is not enough to prove rectangle, rhombus, or square.
5508603
Jamal changes the length of one side of a quadrilateral but keeps the shape closed with the same \(4\) straight sides. He says the shape must now be a pentagon. Is Jamal correct? Explain.

Hints

- Decide which attribute determines the broad polygon category. - Check whether the number of sides changed. - Do not confuse side length with side count.

Solution

1. Changing a side length does not add a side. 2. The shape still has \(4\) straight sides and remains closed. 3. A closed polygon with \(4\) straight sides is still a quadrilateral.

Answer

No. The shape is still a quadrilateral because it still has \(4\) straight sides.
5508613
Look at the three figures. What attribute do all three share that places them in the polygon category?
Figure for problem 550861

Hints

- Do not focus on whether the figures have the same number of sides. - Check the kind of boundary each figure has. - Check whether each boundary closes.

Solution

1. Each figure is closed. 2. Every boundary piece in each figure is a straight segment. 3. Those shared attributes make all three figures polygons even though they have different numbers of sides.

Answer

All three are closed shapes made only of straight sides.
5508623
Priya says, “If two polygons have the same perimeter, they must belong to the same polygon category.” Give one triangle and one quadrilateral with the same perimeter to show that Priya is wrong.

Hints

- Choose a convenient perimeter that can be made with both three and four whole-number side lengths. - Keep the side count different between the two examples. - Check both perimeter sums before using the examples as a counterexample.

Solution

1. A triangle with side lengths \(4, 4, 4\) has perimeter \(12\) units. 2. A quadrilateral with side lengths \(3, 3, 3, 3\) also has perimeter \(12\) units. 3. The shapes have the same perimeter but different numbers of sides, so they belong to different polygon categories.

Answer

One example is a triangle with side lengths \(4, 4, 4\) and a quadrilateral with side lengths \(3, 3, 3, 3\). Both have perimeter \(12\) units.

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