Diketahui kubus ABCD.EFGH dengan rusuk 8 cm. Titik M adalah titik tengah BC. Jarak M ke EG ...

Summary

The video explains how to calculate the distance from point M, the midpoint of BC, to line EG in a cube with an edge length of 8. The solution involves constructing triangle EMG, where EG is the diagonal of the face, and using Pythagoras’ theorem to find lengths of EM and MG. After calculating these lengths, the cosine rule is applied to find the angle at E, followed by the sine rule to determine the final distance. The answer is 6√2 cm, corresponding to option B.

Takeaways

• 📏 The problem involves a cube ABCD EFGH with edge length 8, and M is the midpoint of BC.
• 📐 The goal is to find the distance from M to EG by forming triangle EMG.
• 📊 EG is a diagonal of the face, calculated as 8√2.
• 📈 First, the length of GM is calculated as √(8² + 4²), resulting in 4√5.
• 🧮 To find EM, AM is calculated as √(8² + 4²), giving 4√5 as well.
• 📏 The length of EM is found to be 12 using the Pythagorean theorem.
• 🔢 Using the law of cosines to find angle E in triangle EMG.
• 📝 The cosine rule is applied, yielding cos E as 1/√2.
• 📐 The sine of angle E is then calculated using a right triangle, giving sin E as 1/√2.
• 🎯 Finally, the distance MM' (projection) is computed as 12/√2, which simplifies to 6√2 cm, corresponding to option B.

Q & A

What is the length of the cube's edge in the problem?

The length of the cube's edge is 8 units.

Who is point M in the cube?

Point M is the midpoint of the edge BC of the cube.

What is the goal of the problem?

The goal is to find the distance from point M to the edge EG.

How is the diagonal of the face of the cube EG calculated?

The diagonal EG is calculated as 8√2 units, as it is the diagonal of a square with side length 8.

What formula is used to find the length of MG?

The length of MG is calculated using the Pythagorean theorem as √(8² + 4²), which simplifies to 4√5.

How is the length of EM calculated?

The length of EM is calculated by first finding AM, which is √(8² + 4²) = 4√5, and then using Pythagoras to calculate EM.

What is the length of EM?

The length of EM is 12 units.

Which law is used to find the angle at point E?

The law of cosines is used to find the angle at point E.

What is the cosine of angle E in the triangle EMG?

The cosine of angle E is calculated as (EM² + EG² - MG²) / (2 * EM * EG), which simplifies to 1/√2.

How is the final distance from M to EG calculated?

The final distance from M to EG is calculated using the sine of angle E, resulting in 6√2 cm.