Question
The base of this pyramid is a rectangle. How many congruent faces does this pyramid have?
mc010-1.jpg
The pyramid has 4 congruent faces.
The pyramid has 5 congruent faces.
The pyramid has 1 pair of congruent faces.
The pyramid has 2 pairs of congruent faces.
mc010-1.jpg
The pyramid has 4 congruent faces.
The pyramid has 5 congruent faces.
The pyramid has 1 pair of congruent faces.
The pyramid has 2 pairs of congruent faces.
Answers
GPT-4o mini
To determine how many congruent faces a pyramid with a rectangular base has, let's analyze the structure of such a pyramid:
1. **Base**: The base of the pyramid is a rectangle, which counts as 1 face.
2. **Lateral Faces**: Since the base is a rectangle, there are 4 triangular faces that connect the vertices of the rectangle to the apex of the pyramid.
For the triangular faces:
- The lateral faces can be grouped based on the dimensions of the rectangle.
- If the rectangle's lengths are \( a \) and \( b \), two triangles will have a base \( a \) and the other two triangular faces will have a base \( b \).
We can summarize the faces as follows:
- There is **1 rectangular base face**.
- There are **4 triangular faces**, but since the triangles that share the same base (two triangles with base \( a \) and two with base \( b \)) are congruent to each other respective to their pairs.
Therefore, the 4 triangular faces can be grouped as:
- 2 congruent triangular faces for the base length \( a \).
- 2 congruent triangular faces for the base length \( b \).
All of this leads to the observation that:
- The **pyramid has 5 faces in total** (1 rectangular base + 4 triangular).
- There are **2 pairs of congruent triangular faces**.
Given the response options:
1. The pyramid has **4 congruent faces**. (Incorrect, as there is 1 base and the triangles are in pairs)
2. The pyramid has **5 congruent faces**. (Incorrect, as only triangular faces share congruence)
3. The pyramid has **1 pair of congruent faces**. (Incorrect, as there are 2 pairs)
4. The pyramid has **2 pairs of congruent faces**. (Correct)
Thus, the number of congruent faces the pyramid has is:
**The pyramid has 2 pairs of congruent faces.**
1. **Base**: The base of the pyramid is a rectangle, which counts as 1 face.
2. **Lateral Faces**: Since the base is a rectangle, there are 4 triangular faces that connect the vertices of the rectangle to the apex of the pyramid.
For the triangular faces:
- The lateral faces can be grouped based on the dimensions of the rectangle.
- If the rectangle's lengths are \( a \) and \( b \), two triangles will have a base \( a \) and the other two triangular faces will have a base \( b \).
We can summarize the faces as follows:
- There is **1 rectangular base face**.
- There are **4 triangular faces**, but since the triangles that share the same base (two triangles with base \( a \) and two with base \( b \)) are congruent to each other respective to their pairs.
Therefore, the 4 triangular faces can be grouped as:
- 2 congruent triangular faces for the base length \( a \).
- 2 congruent triangular faces for the base length \( b \).
All of this leads to the observation that:
- The **pyramid has 5 faces in total** (1 rectangular base + 4 triangular).
- There are **2 pairs of congruent triangular faces**.
Given the response options:
1. The pyramid has **4 congruent faces**. (Incorrect, as there is 1 base and the triangles are in pairs)
2. The pyramid has **5 congruent faces**. (Incorrect, as only triangular faces share congruence)
3. The pyramid has **1 pair of congruent faces**. (Incorrect, as there are 2 pairs)
4. The pyramid has **2 pairs of congruent faces**. (Correct)
Thus, the number of congruent faces the pyramid has is:
**The pyramid has 2 pairs of congruent faces.**