To determine the distance from the apex (top) of the right square pyramid to each vertex of the base, follow these steps:
1. Identify the given dimensions:
- Altitude (vertical height) of the pyramid: 10 units
- Each side of the base: 6 units
2. Calculate half of the side length of the base:
- Since the base is a square with each side measuring 6 units, half of the side length is [tex]\( 6 / 2 = 3 \)[/tex] units.
3. Form a right triangle:
- In this right triangle,
- One leg is the altitude (10 units),
- The other leg is half the side of the base (3 units).
4. Apply the Pythagorean theorem to find the slant height (the distance from the apex to each vertex of the base):
- [tex]\( a^2 + b^2 = c^2 \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the legs of the triangle, and [tex]\( c \)[/tex] is the hypotenuse (slant height or distance from the apex to a vertex).
5. Plug in the values:
- Here, [tex]\( a = 10 \)[/tex] (altitude) and [tex]\( b = 3 \)[/tex] (half the side of the base).
6. Calculate the hypotenuse (slant height):
- [tex]\( \text{slant height} = \sqrt{10^2 + 3^2} \)[/tex]
- [tex]\( \sqrt{100 + 9} \)[/tex]
- [tex]\( \sqrt{109} \approx 10.4 \)[/tex]
Thus, the distance from the apex of the pyramid to each vertex of the base is approximately [tex]\( 10.4 \)[/tex] units.
