Question 12

Given a regular pyramid $T.ABCD$ with $AB = 4 \text{ cm}$ and point $O$ is the intersection point of lines $AC$ and $BD$ . What is the volume of the pyramid $T.ABCD$ ?

Regular Pyramid T.ABCD

Visualization of regular pyramid

T.ABCD

with square base

ABCD

and center of base at point

O

Decide whether the following statements $(1)$ and $(2)$ are sufficient to answer the question.

(1) \quad AC = 4\sqrt{2} \text{ cm}

(2) \quad AT = \sqrt{17} \text{ cm}

Statement $(1)$ ALONE is sufficient, but statement $(2)$ ALONE is not sufficient

Statement $(2)$ ALONE is sufficient, but statement $(1)$ ALONE is not sufficient

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

EACH statement ALONE is sufficient

Statements $(1)$ and $(2)$ TOGETHER are NOT sufficient

Explanation

Given a regular pyramid $T.ABCD$ with $AB = 4 \text{ cm}$ and point $O$ is the intersection point of lines $AC$ and $BD$ .

The volume of the pyramid can be calculated using the formula:

V = \frac{1}{3} \times L_{\text{base}} \times h = \frac{1}{3} \times s^2 \times h

where $s$ is the side length of the base and $h$ is the height of the pyramid.

Since $AB = 4 \text{ cm}$ , the base area $L_{\text{base}} = 4^2 = 16 \text{ cm}^2$ .

To calculate the volume, we need to know the height of the pyramid $h = TO$ .

Analysis of Statement $(1)$ : $AC = 4\sqrt{2} \text{ cm}$

The diagonal of a square with side $s$ is $s\sqrt{2}$ .

If $AC = 4\sqrt{2} \text{ cm}$ , then:

s\sqrt{2} = 4\sqrt{2}

s = 4

This is consistent with the given information ( $AB = 4 \text{ cm}$ ).

Point $O$ is the midpoint of the diagonal, so:

AO = OC = \frac{AC}{2} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \text{ cm}

However, statement $(1)$ does not provide information about the height of the pyramid $TO$ . Without the height, the volume cannot be determined.

Therefore, statement $(1)$ ALONE is not sufficient.

Analysis of Statement $(2)$ : $AT = \sqrt{17} \text{ cm}$

Since $AB = 4 \text{ cm}$ , the base diagonal $AC = 4\sqrt{2} \text{ cm}$ .

Point $O$ is the midpoint of the diagonal, so:

AO = \frac{AC}{2} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \text{ cm}

Consider the right triangle $AOT$ with a right angle at $O$ (since $TO$ is perpendicular to the base).

Using the Pythagorean theorem:

AT^2 = AO^2 + TO^2

(\sqrt{17})^2 = (2\sqrt{2})^2 + TO^2

17 = 8 + TO^2

TO^2 = 9

TO = 3 \text{ cm}

With height $TO = 3 \text{ cm}$ and base area $L_{\text{base}} = 16 \text{ cm}^2$ , the volume of the pyramid can be calculated:

V = \frac{1}{3} \times 16 \times 3 = 16 \text{ cm}^3

Therefore, statement $(2)$ ALONE is sufficient to answer the question.

Regular Pyramid T.ABCD with Side Length Labels

Visualization of regular pyramid

T.ABCD

with labels for side lengths.

Conclusion:

Statement $(1)$ ALONE is not sufficient, but statement $(2)$ ALONE is sufficient to answer the question.

Command Palette

Number 12

Explanation