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5209593
Think about a rectangular prism. a) How many vertices and edges does it have? b) Two identical cubes are joined face to face. The seam is inside the new solid and is not an outside edge. What solid is formed, and how many vertices and edges does it have?

Hints

- Count only the vertices and edges on the outside of the solid. - Identify the familiar solid formed by the two cubes.

Solution

1. A rectangular prism has \(8\) vertices and \(12\) edges. 2. Two identical cubes joined face to face form a longer rectangular prism. 3. The new rectangular prism also has \(8\) vertices and \(12\) outside edges.

Answer

a) \(8\) vertices and \(12\) edges b) A rectangular prism with \(8\) vertices and \(12\) edges
5209603
Noah models solids with small wooden balls at the vertices and wire pieces along the edges. a) How many wooden balls and wire pieces does he need for a cube? b) How many wire pieces does he need for a square pyramid? c) Which solid has more vertices, and how many more?

Hints

- Count the vertices and edges of each solid. - A square pyramid has a square base and one top vertex. - Compare the two vertex counts.

Solution

1. A cube has \(8\) vertices and \(12\) edges, so Noah needs \(8\) wooden balls and \(12\) wire pieces. 2. A square pyramid has \(4\) base edges and \(4\) edges from the base to the top vertex, so it has \(4 + 4 = 8\) edges. 3. A square pyramid has \(5\) vertices. Since \(8 - 5 = 3\), the cube has \(3\) more vertices.

Answer

a) \(8\) wooden balls and \(12\) wire pieces b) \(8\) wire pieces c) The cube has \(3\) more vertices.
5372373
The diagram shows rectangle \(ABCD\) and its two diagonals, which intersect at \(M\). Name every triangle in the diagram that has \(\overline{BC}\) as one of its sides.
Figure for problem 537237

Hints

- A triangle is determined by three vertices. Two of them must be \(B\) and \(C\). - Look for other points connected to both \(B\) and \(C\) by drawn segments. - Trace the drawn segments from \(B\) and \(C\) to find each shared third vertex.

Solution

1. Any triangle with side \(\overline{BC}\) must have a third vertex connected to both \(B\) and \(C\). 2. Point \(A\) is connected to \(B\) by a side of the rectangle and to \(C\) by a diagonal, giving \(\triangle ABC\). 3. Point \(D\) is connected to \(B\) by a diagonal and to \(C\) by a side of the rectangle, giving \(\triangle BCD\). 4. Point \(M\) lies on both diagonals and is connected to \(B\) and \(C\), giving \(\triangle BCM\). There are no other labeled points connected to both \(B\) and \(C\).

Answer

The triangles are \(\triangle ABC\), \(\triangle BCD\), and \(\triangle BCM\).

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