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In a regular hexagonal pyramid $SABCDEF$ , the base side $AB = 4$ , and the side edge $SA = 7$ . Point $M$ lies on edge $BC$ , with $BM = 1$ , point $K$ lies on edge $SC$ , with $SK = 4$ . a) Prove that the $MKD$ plane is perpendicular to the plane of the base of the pyramid. b) Find the volume of the pyramid $AB = 4$ $0$ .

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Solve trigonometric equations

Full solution

Q. In a regular hexagonal pyramid

SABCDEF

, the base side

AB = 4

, and the side edge

SA = 7

. Point

M

lies on edge

BC

, with

BM = 1

, point

K

lies on edge

SC

, with

SK = 4

. a) Prove that the

MKD

plane is perpendicular to the plane of the base of the pyramid. b) Find the volume of the pyramid

AB = 4

0

.

Identify Properties: Identify the properties of a regular hexagonal pyramid. In this pyramid, all base sides are equal, and all side edges are equal. The base is a regular hexagon, and the apex is directly above the center of the base.
Calculate Height: Calculate the height of the pyramid using the Pythagorean theorem in triangle SAB, where SA is the slant height, AB/ $2$ is half the base of the hexagon, and SO is the height of the pyramid. Using SA = $7$ and AB = $4$ , we find SO: $\newline$ $SO = \sqrt{SA^2 - (AB/2)^2} = \sqrt{7^2 - (4/2)^2} = \sqrt{49 - 4} = \sqrt{45} \approx 6.71.$
Determine Coordinates: Determine the coordinates for points $M$ and $K$ . Since $M$ is on $BC$ and $BM = 1$ (with $BC = 4$ ), $M$ divides $BC$ in the ratio $1:3$ . Point $K$ is on $K$ $0$ with $K$ $1$ , and $K$ $2$ , so $K$ divides $K$ $0$ in the ratio $K$ $5$ .
Establish Vectors: Establish the vectors for $MK$ and a base edge (like $AB$ ). Calculate the dot product to check if $MK$ is perpendicular to the base. If the dot product is $0$ , $MK$ is perpendicular to the base.

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