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
a^{2} + b^{2} = c^{2}
Example 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.
w^{2} + l^{2} = s^{2}
Step 3 :
Substitute the given measures.
6^{2} + 20^{2} = s^{2}
Simplify.
36 + 400 = s^{2}
436 = s^{2}
Step 3 :
Use the expression for s to find d.
h^{2} + s^{2} = d^{2}
Step 4 :
Plug h = 6 and s^{2} = 436.
6^{2} + 436 = d^{2}
Simplify.
36 + 436 = d^{2}
472 = d^{2}
Take square root on both sides.
√472 = √d^{2}
21.7 ≈ d
Hence, the length of the longest stick that will fit in the box is 21 inches.
Example 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.
w^{2} + l^{2} = s^{2}
Step 3 :
Substitute the given measures.
4^{2} + 14^{2} = s^{2}
Simplify.
16 + 196 = s^{2}
212 = s^{2}
Step 3 :
Use the expression for s to find d.
h^{2} + s^{2} = d^{2}
Step 4 :
Plug h = 4 and s^{2} = 212.
4^{2} + 212 = d^{2}
Simplify.
16 + 212 = d^{2}
228 = d^{2}
Take square root on both sides.
√228 = √d^{2}
15.1 ≈ d
Hence, the greatest length that the part could have been 15 inches.
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