Question:
Select the option that represents the number of triangles in the given description: A regular hexagon whose opposite vertices are all connected by diagonals, forming 6 equilateral triangles meeting at the center, and each of the 6 outer sides has an equilateral triangle constructed externally on it.
उस विकल्प का चयन करें जो दी गई आकृति में त्रिभुजों की संख्या को दर्शाता है: एक सम षट्भुज जिसके सभी विपरीत शीर्ष विकर्णों द्वारा केंद्र पर 6 समबाहु त्रिभुज बनाते हुए जुड़े हैं, और इसके प्रत्येक 6 बाहरी भुजाओं पर बाहर की ओर एक समबाहु त्रिभुज बना है।
Options:
- 20
- 18
- 24
- 28
- 20
- 18
- 24
- 28
Correct Answer: a)
Explanation:
Counting triangles systematically:
1. Single small equilateral triangles:
– 6 inside the hexagon meeting at the center.
– 6 external equilateral triangles constructed on the sides.
Total single units = \(6 + 6 = 12\).
2. Triangles formed by combining 2 or more units:
– Pairs of internal adjacent triangles forming larger triangles through the center? No, 2 equilateral triangles form a rhombus.
– Outer triangle + adjacent internal triangle forms a rhombus.
– Three internal triangles together form a trapezoid.
– Wait, connecting opposite vertices creates large triangles (like hexagram / Star of David formed by 2 large equilateral triangles overlapping):
In a regular hexagon with 3 main diagonals, there are 2 large equilateral triangles formed by alternating vertices. Each large triangle consists of 3 small inner triangles + center parts = 2 triangles.
– Plus combinations with outer points: 6 symmetric medium triangles.
Total systematic count = 20 triangles.
1. Single small equilateral triangles:
– 6 inside the hexagon meeting at the center.
– 6 external equilateral triangles constructed on the sides.
Total single units = \(6 + 6 = 12\).
2. Triangles formed by combining 2 or more units:
– Pairs of internal adjacent triangles forming larger triangles through the center? No, 2 equilateral triangles form a rhombus.
– Outer triangle + adjacent internal triangle forms a rhombus.
– Three internal triangles together form a trapezoid.
– Wait, connecting opposite vertices creates large triangles (like hexagram / Star of David formed by 2 large equilateral triangles overlapping):
In a regular hexagon with 3 main diagonals, there are 2 large equilateral triangles formed by alternating vertices. Each large triangle consists of 3 small inner triangles + center parts = 2 triangles.
– Plus combinations with outer points: 6 symmetric medium triangles.
Total systematic count = 20 triangles.
त्रिभुजों की क्रमबद्ध गणना करने पर:
– 6 आंतरिक एकल त्रिभुज + 6 बाहरी त्रिभुज = 12
– शीर्षों के संयोजन से बनने वाले वृहद समबाहु त्रिभुज एवं सम्मिश्रित त्रिभुज = 8
कुल त्रिभुजों की संख्या = 20।
– 6 आंतरिक एकल त्रिभुज + 6 बाहरी त्रिभुज = 12
– शीर्षों के संयोजन से बनने वाले वृहद समबाहु त्रिभुज एवं सम्मिश्रित त्रिभुज = 8
कुल त्रिभुजों की संख्या = 20।