Jarak Titik E ke Garis BP pada Kubus

Hello everyone! Today we are calculating the distance from a point to a line in a cube. We have cube A B C D E F G H with a side length of twelve centimeters. Point P is the midpoint of edge H G, and our task is to determine the distance from point E to the line B P.

Step one is to calculate the lengths of the triangle sides. E B is a face diagonal, equal to twelve square root of two centimeters. E P is calculated as the square root of E H squared plus H P squared, which is the square root of twelve squared plus six squared, resulting in six square root of five. B P is the square root of B C squared plus C G squared plus G P squared, which simplifies to the square root of twelve squared plus twelve squared plus six squared, equaling eighteen centimeters.

Step two is to find the cosine of angle B and then the distance. Using the law of cosines, cosine B equals E B squared plus B P squared minus E P squared, divided by two times E B times B P. This gives us two hundred eighty-eight plus three hundred twenty-four minus one hundred eighty, all over four hundred thirty-two square root of two. This simplifies to four hundred thirty-two divided by four hundred thirty-two square root of two, which is one over the square root of two. This means angle B equals forty-five degrees.

Now we find the distance d, which equals E B times the sine of angle B. Substituting our values, d equals twelve square root of two times the sine of forty-five degrees. This becomes twelve square root of two times one half square root of two.

The final calculation gives us a distance of exactly twelve centimeters. And that is our final answer!