**Pythagorean theorem in three dimensions :**

We can can use the Pythagorean Theorem to solve real-world problems in three dimensions.

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

If a and b are legs and c is the hypotenuse, then

a² + b² = c²

**Problem 1 : **

A box used for shipping narrow silver sticks measures 6 inches by 6 inches by 20 inches. What is the length of the longest stick that will fit in the box, given that the length of the tube must be a whole number of inches ?

**Solution : **

**Step 1 : **

Draw an appropriate diagram for the given information.

From the diagram given above, the box has the following dimensions.

Length (l) = 20 in.

Width (w) = 6 in.

Height (h) = 6 in.

**Step 2 :**

We want to find d, the length from a bottom corner to the opposite top corner. First, find s, the length of the diagonal across the bottom of the box.

w² + l² = s²

**Step 3 :**

Substitute the given measures.

6² + 20² = s²

Simplify.

36 + 400 = s²

436 = s²

**Step 3 :**

Use the expression for s to find d.

h² + s² = d²

**Step 4 :**

Plug h = 6 and s² = 436.

6² + 436 = d²

Simplify.

36 + 436 = d²

472 = d²

Take square root on both sides.

√472 = √d²

21.7 ≈ d

Hence, the length of the longest stick that will fit in the box is 21 inches.

**Problem 2 : **

Lily ordered a replacement part for her desk. It was shipped in a box that measures 4 in. by 4 in. by 14 in. What is the greatest length in whole inches that the part could have been ?

**Solution : **

**Step 1 : **

Draw an appropriate diagram for the given information.

From the diagram given above, the box has the following dimensions.

Length (l) = 14 in.

Width (w) = 4 in.

Height (h) = 4 in.

**Step 2 :**

We want to find d, the length from a bottom corner to the opposite top corner. First, find s, the length of the diagonal across the bottom of the box.

w² + l² = s²

**Step 3 :**

Substitute the given measures.

4² + 14² = s²

Simplify.

16 + 196 = s²

212 = s²

**Step 3 :**

Use the expression for s to find d.

h² + s² = d²

**Step 4 :**

Plug h = 4 and s² = 212.

4² + 212 = d²

Simplify.

16 + 212 = d²

228 = d²

Take square root on both sides.

√228 = √d²

15.1 ≈ d

Hence, the greatest length that the part could have been 15 inches.

After having gone through the stuff given above, we hope that the students would have understood "Pythagorean theorem in three dimensions".

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