Math Problem Statement

give me answer

ABCDEF is a cuboid. DE = 12cm, BE = 13cm, EF = 3cm, calculate the length of FG.

Solution

The cuboid in the image has the following dimensions:

  • DE=12cmDE = 12 \, \text{cm}
  • BE=13cmBE = 13 \, \text{cm}
  • EF=3cmEF = 3 \, \text{cm}

To calculate the length of FGFG, we need to recognize that FGFG is a diagonal in the base of the cuboid formed by the vertices E,F,G,HE, F, G, H.

Step 1: Recognize the triangle

The points F,G,EF, G, E form a right-angled triangle:

  • EF=3cmEF = 3 \, \text{cm} (given),
  • FGFG is what we need to calculate,
  • EGEG is the diagonal of rectangle EFGHEFGH, where EH=DE=12cmEH = DE = 12 \, \text{cm}.

Step 2: Calculate diagonal EGEG

EGEG is the diagonal of a rectangle with sides EFEF and EHEH. Using the Pythagorean theorem for right triangle EFGEFG:

EG=EH2+EF2EG = \sqrt{EH^2 + EF^2}

Substitute the values:

EG=122+32=144+9=15312.37cmEG = \sqrt{12^2 + 3^2} = \sqrt{144 + 9} = \sqrt{153} \approx 12.37 \, \text{cm}

Step 3: Use Pythagorean theorem for triangle FEGFEG

Now that we know EGEG, we can find FGFG. We apply the Pythagorean theorem again, considering the right triangle FEGFEG:

FG2=EG2+BE2FG^2 = EG^2 + BE^2

Substitute the values:

FG2=12.372+132=153+169=322FG^2 = 12.37^2 + 13^2 = 153 + 169 = 322

Finally, calculate FGFG:

FG=32217.94cmFG = \sqrt{322} \approx 17.94 \, \text{cm}

Final Answer:

The length of FGFG is approximately 17.94 cm.


Would you like more details or explanations on any steps?

Here are 5 related questions to explore:

  1. How would you calculate the diagonal of a cuboid?
  2. Can the Pythagorean theorem be applied in three dimensions?
  3. How would you calculate the surface area of this cuboid?
  4. What is the formula for the space diagonal of a cuboid?
  5. How can trigonometry be used to solve cuboid problems?

Tip: Always check if you're working with a right-angled triangle when applying the Pythagorean theorem, as it simplifies calculations.

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Math Problem Analysis

Mathematical Concepts

Geometry
3D Shapes
Pythagorean Theorem

Formulas

Diagonal of a rectangle: d = √(a² + b²)
Pythagorean theorem: c² = a² + b²

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 9-10