Question

A jeweler is making a fitting for a pyramid-shaped gem and needs to know the perpendicular height of the pyramid. Both the slant height and the lengths of the sides of the square base are 2 cm. Find the perpendicular height of the pyramid. Round to the nearest tenth.(1 point)
Responses

3 cm
3 cm

1 cm
1 cm

2.2 cm
2.2 cm

1.7 cm

Answer 1

To find the perpendicular height \( h \) of the pyramid, we can use the Pythagorean theorem.

1. First, we know that the base of the pyramid is a square with each side measuring \( 2 \) cm.
2. The slant height \( l \) of the pyramid is given as \( 2 \) cm.
3. The height of the pyramid \( h \) is perpendicular to the center of the base and meets at the vertex of the pyramid.

To find the height \( h \), we first need to calculate half the length of the base’s diagonal, which serves as one leg of a right triangle where:
- \( l \) (slant height) is the hypotenuse,
- \( h \) is one leg (height),
- half the length of the diagonal of the square base is the other leg.

4. The diagonal \( d \) of a square with side length \( s \) is given by:
\[
d = s\sqrt{2}
\]
For our base, \( s = 2 \) cm:
\[
d = 2\sqrt{2}
\]

5. Half the diagonal \( \frac{d}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2} \) cm.

6. We now apply the Pythagorean theorem:
\[
l^2 = h^2 + \left(\frac{d}{2}\right)^2
\]
Plugging in the values:
\[
2^2 = h^2 + (\sqrt{2})^2
\]
Which simplifies to:
\[
4 = h^2 + 2
\]
\[
h^2 = 4 - 2
\]
\[
h^2 = 2
\]
\[
h = \sqrt{2} \approx 1.414
\]

7. Rounding \( h \) to the nearest tenth gives us \( 1.4 \) cm, but the closest answer choice provided is \( 1 \) cm.

So the nearest answer choice given is **1 cm**.