Question

consider the following pyramids, whose bases are a square and an equilateral triangle.
kyra tried to use cavalieris principle to show that the two figures have the same volume.
\the base areas are the same. therefore, corresponding cross - sections have the same area. so the volumes must be the same.\
what is the first mistake kyra made?
choose 1 answer:
a the base areas are not the same.
b it isnt true that corresponding cross - sections have the same area.
c she did not establish that the heights are the same.

Explanation:

To determine Kyra's first mistake, we analyze each option:

• Option A: The base of the first pyramid is a square with side 3, so its area is \(3\times3 = 9\). The base of the second pyramid is an equilateral triangle with side 3. The area of an equilateral triangle is \(\frac{\sqrt{3}}{4}s^2=\frac{\sqrt{3}}{4}\times3^2=\frac{9\sqrt{3}}{4}\approx3.897\), which is not equal to 9. Wait, no—wait, actually, let's re - check. Wait, the first base is a square: area \(A_{square}=3\times3 = 9\). The second base is an equilateral triangle: area \(A_{triangle}=\frac{\sqrt{3}}{4}\times3^2=\frac{9\sqrt{3}}{4}\approx3.897\). But wait, the problem is about Kyra's reasoning. Wait, no, the key here is Cavalieri's principle. Cavalieri's principle states that if two solids have the same height and the same cross - sectional area at every level, then they have the same volume. But first, we need to check the base areas and the heights. Wait, no, let's re - evaluate the options. Wait, the first mistake: Kyra said "the base areas are the same". Let's calculate the base areas. Square base: side length 3, area \(3\times3 = 9\). Equilateral triangle base: side length 3, area \(\frac{\sqrt{3}}{4}\times3^{2}=\frac{9\sqrt{3}}{4}\approx3.897\). So the base areas are not the same. But wait, no—wait, maybe I made a mistake. Wait, no, the first option is "The base areas are not the same". But let's check the other options. Option C: She did not establish that the heights are the same. But the slant height is 11, but the height (the perpendicular height) of the pyramid: for the square - based pyramid, the slant edge is 11? Wait, no, the diagram shows the slant edge? Wait, maybe the height (the perpendicular height from the apex to the base) is the same? Wait, no, the problem is about Kyra's first mistake. Let's re - read Kyra's statement: "The base areas are the same. Therefore, corresponding cross - sections have the same area. So the volumes must be the same." The first part of her statement is wrong because the base areas are not the same. The square base has area \(3\times3 = 9\), the equilateral triangle base has area \(\frac{\sqrt{3}}{4}\times3^{2}=\frac{9\sqrt{3}}{4}\approx3.897\), so they are not the same. So the first mistake is that the base areas are not the same.

Answer:

A. The base areas are not the same.