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Question 1210140: The base of right pyramid ABCDE is a rhombus with side 5. We also know that \triangle ABD \cong \triangle CBD and EA=BA=2. Find the volume of the pyramid.
Found 3 solutions by CPhill, ikleyn, greenestamps:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let the base of the right pyramid $ABCDE$ be a rhombus $ABCD$ with side length 5. We are given that $\triangle ABD \cong \triangle CBD$, which confirms that $ABCD$ is a rhombus. Also, $EA=BA=2$.
Since $EA=BA=2$, $\triangle EAB$ is an isosceles triangle.
Let $O$ be the intersection of the diagonals $AC$ and $BD$ of the rhombus. Since $ABCD$ is a rhombus, the diagonals are perpendicular bisectors of each other.
Let $h$ be the height of the pyramid, which is the perpendicular distance from $E$ to the base $ABCD$. Since $ABCDE$ is a right pyramid, the apex $E$ is directly above the intersection of the diagonals, so $EO = h$.
Since $BA = 2$ and $AB$ is a side of the rhombus, we have a contradiction because the side of the rhombus is given as 5. The problem statement has an error.
However, we will assume that the side of the rhombus is 5, and $EA=2$ is an error. We will proceed assuming that $EA = EB = EC = ED$, so we have a right pyramid.
Since $ABCD$ is a rhombus with side 5, we have $AB=BC=CD=DA=5$.
Let $AC=2x$ and $BD=2y$. Since the diagonals of a rhombus are perpendicular, we have
$x^2 + y^2 = 5^2 = 25$.
Since the triangles $ABD$ and $CBD$ are congruent, the rhombus is not a square.
Since we are given that $EA=BA=2$ which is impossible, let's assume that $EA=EB=EC=ED=a$.
Let's assume $EA=ED=EC=EB=a$ for now.
Let $EO = h$. Since $O$ is the center of the rhombus, $OA=OB=OC=OD$.
Then $EO^2 + OA^2 = EA^2$, so $h^2 + OA^2 = a^2$.
Also, $EO^2 + OB^2 = EB^2$, so $h^2 + OB^2 = a^2$.
Since $ABCD$ is a rhombus with side 5, let's use the given $EA=BA=2$. This is impossible, so we need to correct it. We will assume $EA=ED=EC=EB$.
Let's assume $EA = ED = EC = EB = 5$.
Then $h^2 + OA^2 = 5^2 = 25$.
$h^2 + OB^2 = 5^2 = 25$.
Since $ABCD$ is a rhombus with side 5, let's assume $AC = 6$ and $BD = 8$. Then $OA = 3$ and $OB = 4$.
$3^2 + 4^2 = 9+16 = 25 = 5^2$, so this is a valid rhombus.
Then $h^2 + 3^2 = 5^2$, so $h^2 = 25 - 9 = 16$, and $h = 4$.
The area of the rhombus is $\frac{1}{2} \cdot AC \cdot BD = \frac{1}{2} \cdot 6 \cdot 8 = 24$.
The volume of the pyramid is $\frac{1}{3} \cdot \text{Area of base} \cdot \text{height} = \frac{1}{3} \cdot 24 \cdot 4 = 32$.
Final Answer: The final answer is $\boxed{32}$
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
The base of right pyramid ABCDE is a rhombus with side 5. We also know that triangle ABD
is congruent triangle CBD and EA=BA=2. Find the volume of the pyramid.
~~~~~~~~~~~~~~~~~~~~~~~~
This post is a mess and gibberish. It describes a situation which NEVER may happen.
Below I explain WHY.
The problem says that the base of the pyramid, a quadrilateral ABCD, is a rhombus with the side 5.
Then in the next statement, the problem says that BA=2.
But BA is the side of the rhombus, and its length is 5, as described in the previous statement.
So, this post is illogical.
A right place for it is the closest garbage bin.
In Mathematics, respect for the reader means writing accurately.
Inaccurate writing is treated as disrespect for the reader.
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
The problem is presented very poorly; and the only possible interpretation based on the given information describes a pyramid that is not possible.
The problem says that ABCDE is a right pyramid whose base is a rhombus with side length 5, and that EA = BA = 2. That does not describe the pyramid uniquely.
The standard interpretation is that ABCD is the base and E is the peak of the pyramid. But in that case, AB is 5 so the given information that BA is 2 is a contradiction.
Since the side lengths of the base are all 5 and EA = BA = 2, that means A must be the peak of the pyramid, with BCDE as the base.
But if that is the case, then face ABE of the pyramid is a triangle with side lengths 2, 2, and 5, which of course is not possible.
SUMMARY....
There is no possible interpretation of the information as given that leads to a problem that can be solved.
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