Rectangular Parallelepiped Problem, 12

Category: Solid Geometry, Plane Geometry

"Published in Newark, California, USA"

In the figure is shown a rectangular parallelepiped whose dimensions are 2, 4, 6. Points A, B, C, E, F, and L are each at the midpoint of an edge. Find the area of each of the sections ABEF, ABC, and MNL.

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Solution:

If points A, B, C, E, F, and L are the midpoint of the edges of a rectangular parallelepiped, then we can label further the figure above as follows

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By Pythagorean Theorem, the length of AB is

By Pythagorean Theorem, the length of BC is

By Pythagorean Theorem, the length of AC is

Parallelogram ABEF is a rectangle because AB ║ EF , BE ║ AF, AB ≅ EF, and BE ≅ AF. If AB = 3.1623, then it follows that EF = 3.1623.

By Pythagorean Theorem, the length of MN is

By Pythagorean Theorem, the length of NL is

By Pythagorean Theorem, the length of ML is

We notice that ∆MNL is an isosceles triangle because MN
≅ ML = 6.3246. 

Therefore, the area of rectangle ABEF is

For ∆ABC, the semi-perimeter is

Therefore, the area of ∆ABC by using Heron's Formula is

For ∆MNL, the semi-perimeter is

Therefore, the area of ∆MNL by using Heron's Formula is

Rectangular Parallelepiped Problem, 11

Category: Solid Geometry, Plane Geometry

"Published in Vacaville, California, USA"

If a, b, and c are the sides of a rectangular parallelepiped, find the length of its diagonal.

Solution:

To illustrate the problem, it is better to draw the figure as follows

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The first thing that we need to do is to consider the lower base of the above figure first and then solve for the length of the diagonal by using Pythagorean Theorem.

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By Pythagorean Theorem, the length of a diagonal of the lower base is

Since all sides as well as the faces of a rectangular parallelepiped are perpendicular to each other, then it follows that x is perpendicular to c. 

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Therefore, by Pythagorean Theorem, the length of a diagonal of a rectangular parallelepiped is

The diagonal of a rectangular parallelepiped is a length or a distance between the opposite vertices. You can draw four diagonals in a rectangular parallelepiped.

Rectangular Parallelepiped Problem, 10

Category: Solid Geometry

"Published in Vacaville, California, USA"

A solid concrete porch consists of 3 steps and a landing. The steps have a tread of 11 in., a rise of 7 in., and a length of ft.; the landing is 6 ft. by 7 ft. How much material was used in its construction?

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Solution:

Consider the given figure above

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The first that we need to do is to divide the given figure into rectangular parallelepiped as follows

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There are one large rectangular parallelepiped and 6 small rectangular parallelepiped in the given figure. Let's assume that the steps of a ladder have 6 small rectangular parallelepiped whose dimensions are all congruent to each other. 

For the large rectangular parallelepiped which is the landing, the volume is

For the small rectangular parallelepiped which is the step of a ladder, the volume is

Therefore, the volume of a ladder which is the amount of a material used in its construction is equal to