Math

QuestionGegeben sind die Punkte A(2,-3,0), B(2,2,0), C(-1,2,0) und E(2,-3,5) eines Quaders. Finde die anderen Eckpunkte und berechne die Diagonale AG\overline{A G}.

Studdy Solution

STEP 1

Assumptions1. The given points A(|-3|0), B(||0), C(-1||0), and(|-3|5) are vertices of a cuboid. . A, B, and C are vertices on the base of the cuboid and is a vertex on the top face of the cuboid.
3. The cuboid is aligned with the axes of the coordinate system.

STEP 2

First, let's find the coordinates of the other vertices of the cuboid. Since A, B, and C are on the base, the corresponding vertices on the top face will have the same x and y coordinates, but the z coordinate will be the same as.

STEP 3

The coordinates of the vertex corresponding to A on the top face are (2, -3,5).

STEP 4

The coordinates of the vertex corresponding to B on the top face are (2,2,).

STEP 5

The coordinates of the vertex corresponding to C on the top face are (-1,2,5).

STEP 6

Now, let's find the coordinates of the vertex on the base that corresponds to. Since is on the top face, the corresponding vertex on the base will have the same x and y coordinates, but the z coordinate will be0.

STEP 7

The coordinates of the vertex corresponding to on the base are (2, -3,0). But we already have this vertex, it's A. So, we have found all the vertices of the cuboid.

STEP 8

Now, let's calculate the length of the space diagonal AG. The space diagonal of a cuboid is the line segment connecting opposite vertices of the cuboid. In this case, A is one of the vertices on the base and G is the corresponding vertex on the top face.

STEP 9

The length of a line segment in three dimensions can be calculated using the distance formulad=(x2x)2+(y2y)2+(z2z)2d = \sqrt{(x2-x)^2 + (y2-y)^2 + (z2-z)^2}

STEP 10

Plug in the coordinates of A and G into the distance formula to calculate the length of AG.
d=(22)2+(3+3)2+(05)2d = \sqrt{(2-2)^2 + (-3+3)^2 + (0-5)^2}

STEP 11

Calculate the length of AG.
d = \sqrt{(0)^ + (0)^ + (-5)^} = \sqrt{25} =5The length of the space diagonal AG is5.

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