SOLUTION: VABC is a right pyramid whose base is an equilateral triangle Abc of side 8cm. If VA = 13cm and the altitude of DABC meet at x. Calculate AX, VX, the angle which VAB makes with A

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Question 1198890: VABC is a right pyramid whose base is an equilateral triangle Abc of side 8cm. If VA = 13cm and the altitude of DABC meet at x. Calculate
AX, VX, the angle which VAB makes with ABC
Answer by textot(100) About Me  (Show Source):
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**1. Find the Height of the Equilateral Triangle Base (Altitude)**
* In an equilateral triangle, the altitude bisects the base.
* Let the altitude of the equilateral triangle be 'h'.
* Using the Pythagorean theorem in one of the right-angled triangles formed by the altitude:
(h)² + (4)² = (8)²
h² = 64 - 16 = 48
h = √48 = 4√3 cm
**2. Find the Height of the Pyramid (VX)**
* Consider the right triangle VAX, where VX is the height of the pyramid.
* Using the Pythagorean theorem:
(VX)² + (AX)² = (VA)²
VX² + (4√3)² = (13)²
VX² + 48 = 169
VX² = 121
VX = 11 cm
**3. Find AX**
* From the previous step, we found that AX = 4√3 cm.
**4. Find the Angle VAB**
* Consider the right triangle VAX.
* Let the angle VAB be θ.
* cos(θ) = AX / VA
* cos(θ) = (4√3) / 13
* θ = arccos((4√3) / 13)
* θ ≈ 49.11 degrees
**Therefore:**
* **AX = 4√3 cm**
* **VX = 11 cm**
* **Angle VAB ≈ 49.11 degrees**