The base of a solid oblique pyramid is an equilateral triangle with a base edge length of 14 units.

A solid oblique pyramid has an equilateral triangle base with a base edge length of 14 units. The base triangle is triangle A C D. The apex is point B. The angles formed with the lateral sides are 45 degrees.

What is BC, the height of the pyramid?

7 units
7StartRoot 2 EndRoot units
14 units
14StartRoot 2 EndRoot units

To find the height \( BC \) of the oblique pyramid, we first need to understand the geometry of the pyramid. 

1. **Base Triangle Calculation**:
   The base of the pyramid is an equilateral triangle \( \triangle ACD \) with side length 14 units.

2. **Properties of an Equilateral Triangle**:
   For an equilateral triangle, the altitude can be found using the formula:
   \[
   \text{altitude} = \frac{\sqrt{3}}{2} \times \text{side length}
   \]
   So, the altitude of \( \triangle ACD \) is:
   \[
   \text{altitude} = \frac{\sqrt{3}}{2} \times 14 = 7\sqrt{3} \text{ units}
   \]
   This altitude runs from a vertex of the triangle to the midpoint of the opposite side.

3. **Calculating the Height \( BC \)**:
   The lateral faces form a \( 45^\circ \) angle with the base. To find the height \( BC \), we will use the trigonometric relationship:
   \[
   \tan(45^\circ) = \frac{\text{altitude of the base triangle}}{\text{height of the pyramid}}
   \]
   Since \( \tan(45^\circ) = 1 \), we have:
   \[
   1 = \frac{7\sqrt{3}}{BC}
   \]
   Solving for \( BC \), we get:
   \[
   BC = 7\sqrt{3}
   \]

4. **Simplifying \( 7\sqrt{3} \)**:
   If we check the given answer choices, none matches \( 7\sqrt{3} \) exactly. This indicates a need to revisit the alignment or intended structure.

Thus, the answer closest is indeed \(14

The base of a solid oblique pyramid is an equilateral triangle with a base edge length of 14 units. The apex is point B, and the angles formed with the lateral sides are 45 degrees. What is BC, the height of the pyramid?

Discover how to calculate the height of an oblique pyramid with an equilateral triangle base of side length 14 units, using trigonometry and geometric principles. This solution explains how to find BC, the height, based on a 45-degree angle. Suitable for students in Grades 9-12.

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