Question

What is the longest diagonal of a rectangular box that is $120$ inches long, $90$ inches wide and $80$ inches tall?
A)$170$ B)$120$ C)$150$ D)$190$

Answer 1

Hint: To find the length of longest diagonal of a three-dimensional rectangular box use the Pythagoras theorem twice which is ${{\text{H}}^2}{\text{ = }}{{\text{P}}^2}{\text{ + }}{{\text{B}}^2}$ where H is hypotenuse, P is perpendicular and B is the base of triangle.

Complete step-by-step answer:
We are given that a rectangular box is $120$ inches long, $90$ inches wide and $80$ inches tall. We have to find its longest diagonal. Since rectangular box is a three dimensional structure, we can draw it as-

Here a represents the diagonal at the bottom of the box and x represents the longest diagonal. First we have to find a. So in the structure we can see that at the bottom of the box the diagonal forms a right angled triangle with the sides of the rectangle. So we can use Pythagoras theorem to find ‘a’ as we know the value of the base and perpendicular of the triangle. According to the Pythagoras theorem, “In a right angled triangle, the square of hypotenuse is equal to the sum of squares of other sides of the triangle.”
$ \Rightarrow {{\text{H}}^2}{\text{ = }}{{\text{P}}^2}{\text{ + }}{{\text{B}}^2}$ Where H is hypotenuse, P is perpendicular and B is the base of triangle.
From the figure, it is clear that H=a, P=$120$inches and B=$90$inches. On putting these values in formula, we get-
$ \Rightarrow {\left( {\text{a}} \right)^2} = {\left( {120} \right)^2} + {\left( {90} \right)^2}$
On simplifying we get,
$ \Rightarrow {\left( {\text{a}} \right)^2} = 14400 + 8100 = 22500$
On removing the square-root from a, we get-
$ \Rightarrow {\text{a}} = \sqrt {22500} = 150$ Inches
Now we can find the value of x. From the figure it is clear that diagonal x forms a right angled triangle with ‘a’ and other sides of the rectangle. So we can use Pythagoras theorem to find ‘x’ as we know the value of the base and perpendicular of the triangle. According to the Pythagoras theorem, “In a right angled triangle, the square of hypotenuse is equal to the sum of squares of other sides of the triangle.”
$ \Rightarrow {{\text{H}}^2}{\text{ = }}{{\text{P}}^2}{\text{ + }}{{\text{B}}^2}$ Where H is hypotenuse, P is perpendicular and B is the base of triangle.
Here, H=x, P=$80$ and B=$150$ =a. On putting these values in the formula we get,
$ \Rightarrow {\left( {\text{x}} \right)^2} = {\left( {150} \right)^2} + {\left( {80} \right)^2}$
On simplifying, we get-
$ \Rightarrow {\left( {\text{x}} \right)^2} = 22500 + 6400 = 28900$
On removing square-root from x, we get-
$ \Rightarrow {\text{x = }}\sqrt {28900} = 170$ inches
Hence the answer is ‘A’.
Note: You can also solve this question by using Deluxe Pythagoras theorem which is used for finding the diagonal in three dimensional solids. Deluxe Pythagoras theorem is given as-
$ \Rightarrow $ \[{\left( {{\text{Length of diagonal}}} \right)^2}\] =${{\text{x}}^2} + {{\text{y}}^2} + {{\text{c}}^2}$ where x, y, and z are given sides of rectangle.
On putting the given values of question we get,
$ \Rightarrow $ \[{\left( {{\text{Length of diagonal}}} \right)^2}\]=${\left( {120} \right)^2} + {\left( {90} \right)^2} + {\left( {80} \right)^2} = 1440 + 8100 + 6400 = 28900$
$ \Rightarrow $ \[\left( {{\text{Length of diagonal}}} \right) = \sqrt {28900} = 170\]inches
This is a short method to find the answer.