Triangle and Trapezoid Problems, 2

Category: Plane Geometry

"Published in Vacaville, California, USA"

Find the area of the shaded region

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Solution:

Consider the given figure above

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The small triangle is an isosceles triangle because the two adjacent sides are congruent. Same thing with the big triangle. If the bases of two triangles are parallel with the same or common vertex, then two triangles are similar. 

Since the two bases of the shaded region is parallel and the opposite sides are congruent, then the figure of the shaded region is an isosceles trapezoid. 

The length of the lower base as well as the altitude of an isosceles trapezoid are unknown, then we need to label further the figure first as follows

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The altitude of big triangle bisects the bases of the small and big triangles with their same or common vertex.

By using ratio and proportion since two triangles are similar, the length of the lower base of an isosceles trapezoid is

By Pythagorean Theorem, the altitude of the small triangle is

By using ratio and proportion since two triangles are similar, the altitude of an isosceles trapezoid is 

Therefore, the area of the shaded region which is an isosceles trapezoid is

                               or

 

Triangle and Trapezoid Problems

Category: Plane Geometry

"Published in Newark, California, USA"

Find the area of the given figure

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above 
 

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The given figure consists of equilateral triangle and a right trapezoid. The altitude of a right trapezoid is also a side of an equilateral triangle. Since the altitude of an equilateral triangle is not given, then we have to solve it first as follows

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By Pythagorean Theorem, the altitude of an equilateral triangle is

 

 

 

 

The area of an equilateral triangle is

The area of a right trapezoid is

 

 

Therefore, the area of a given figure is
 

 

 

                             or

 

Circle Inscribed in a Trapezoid Problems

Category: Plane Geometry

"Published in Newark, California, USA"

Find the exact area of the given trapezoid in the figure:

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Solution:

Consider the given figure above

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Since the given figure is an isosceles trapezoid, then it follows that ∠A ≅ ∠B, ∠C ≅ ∠D, and AD ≅ BC. If a circle is inscribed in an isosceles trapezoid, then its radius is tangent to the sides of an isosceles trapezoid. Let's analyze and label further the given figure as follows
 

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Line segment EG that passes the center of a circle bisects the two bases of an isosceles trapezoid. Line segment OB bisects ∠B and line segment OC bisects ∠C. 

Consider rt. ∆OGB and rt. ∆OFG. If OG ≅ OF and OB ≅ OB, then it follows that BG ≅ BF. In this case, BF = 9.

Consider rt. ∆OEC and rt. ∆OFC. If OE ≅ OF and OC ≅ OC, then it follows that EC ≅ CF. In this case, CF = 4.

Hence, the length of CB = CF + FB = 9 + 4 = 13.

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The value of x is 

By Pythagorean Theorem, the altitude of an isosceles trapezoid is

 

 

 

 

Therefore, the area of an isosceles trapezoid is