More Cube Problem, 7

Category: Solid Geometry, Trigonometry

"Published in Vacaville, California, USA"

The plane section ABCD shown in the figure is cut from a cube of edge a. Find the angle which the section ABCD makes with the lower base of the cube if D and C are each at the midpoint of an edge.

Photo by Math Principles in Everyday Life

Solution:

To understand more the problem, it is better to label further the given figure as follows

Photo by Math Principles in Everyday Life

Since the edges of a cutting plane which are AB and CD are parallel to the four parallel sides of a cube, then two triangles formed by a cutting plane with a cube are congruent. 

Since the edges of a cube are perpendicular to each other, then two triangles are right triangles.

Therefore, the angle of a cutting plane with respect to the lower base of a cube is

 

or

 

More Triangle Problems, 6

Category: Plane Geometry, Trigonometry

"Published in Newark, California, USA"

Show that, for the two triangles ABC and AXY (see figure),

Photo by Math Principles in Everyday Life

Solution:

Since the altitude of the two triangles are not given, then we can draw the altitudes from points B and X as follows

Photo by Math Principles in Everyday Life

Consider ∆ABC:

but

then the above equation becomes

Consider ∆AXY:

but

then the above equation becomes

Therefore,

Trapezoid - Circular Segment Problems, 2

Category: Plane Geometry, Trigonometry

"Published in Vacaville, California, USA"

The plane area shown in the figure consists of an isosceles trapezoid (non-parallel sides equal) and a segment of a circle. If the non-parallel sides are tangent to the segment at points A and B, find the area of the composite figure.

Photo by Math Principles in Everyday Life

Solution:

The given plane figure consists of an isosceles trapezoid and a circular segment. Let's analyze and label further the above figure as follows

Photo by Math Principles in Everyday Life

From point A, draw a line perpendicular to CA and from point B, draw a line perpendicular to BD. The intersection of the two lines is point O which is the center of a circular arc. By using the laws of angles, ∆AOB is an isosceles triangle because the two base angles and their opposite sides are  equal. If ∆AOB is an isosceles triangle, then the altitude h bisects AB into two equal parts which are 1.5'' each.

The radius of a circular arc is

The height of ∆AOB is

The angle of a circular arc is

Hence, the area of a circular segment is 

The height of a circular segment is

The height of a trapezoid is

The length of the upper base of a trapezoid is

but

Then the above equation becomes

Hence, the area of a trapezoid is 

Therefore, the area of a plane figure is