Square, Rectangle, and Parallelogram Problems, 15

Category: Plane Geometry

"Published in Vacaville, California, USA"

A rectangular parcel of land is 50 ft wide. The length of a diagonal between opposite corners is 10 ft more than the length of the parcel. What is the length of the parcel?

Solution:

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

Photo by Math Principles in Everyday Life

If a rectangle is cut by its diagonal, then the resulting figure is a pair of right triangles. By Pythagorean Theorem, we can solve for the value of x as follows

Therefore, the length of a rectangular parcel of land is 120 ft.

Square, Rectangle, and Parallelogram Problems, 14

Category: Plane Geometry

"Published in Vacaville, California, USA"

A rectangular parcel of land has an area of 5000 ft². A diagonal between opposite corners is measured to be 10 ft longer than one side of the parcel. What are the dimensions of the land, correct to the nearest foot?

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

If a rectangle is cut by its diagonal, then the resulting figure is a pair of right triangles. By Pythagorean Theorem, the width of a rectangle is

Hence, the area of a rectangle is 

Square on both sides of the equation, we have

Since the resulting equation cannot be factored by synthetic division, factoring, grouping, or substitution method, then we have to do the trial and error method by assigning the values of x until the left side of the equation equals the right side of the equation which is zero as follows

Photo by Math Principles in Everyday Life

It is better to do the calculations if you will use Microsoft Excel because you can put the formula like for instance, the resulting cubic equation that we can't factor by any method. You can enter any values of x and the Excel program will give us the result as shown at the table above.

From the result as shown at the table above, we have to choose the value of x which is 106.0805885 because the result is closer to zero. Hence, the length of a rectangle is 106.0805885 ft ≈ 106 ft. 

The width of a rectangle is

Therefore, the dimensions of the rectangular land are 106 ft by 47 ft. 

Three Tangent Circles, 2

Category: Plane Geometry

"Published in Vacaville, California, USA"

Three tangent circles of radius 10 cm are drawn. All centers lie on the line AB. The tangent AC to the right-hand circle is drawn, intersecting the middle circle at D and E. Find the length of the segment DE.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

Since AC is tangent to the third circle at C, then the radius of third circle is perpendicular to AC. Hence, the three lines becomes a right triangle as follows

Photo by Math Principles in Everyday Life

By Pythagorean Theorem, the length of AC is

At the center of second circle, draw a line which is perpendicular to DE. Extend that line at the opposite ends of second circle as follows

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If SQ is perpendicular to DE and passes thru the center of the circle P, then SQ bisects DE. Since RP is perpendicular to AC and CO is perpendicular to AC, then RP is parallel to CO. Because of this ∆ARP and ∆ACO are similar.

If ∆ARP and ∆ACO are similar, then we can solve for RP by ratio and proportion as follows

If a theorem says "When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.", then the working equation for the second circle is

Therefore, the length of DE is