(ii) 6
2. A regular polygon has 7 sides. Find
(a) The exterior angle
(b) The interior angle
3. The interior angles of a pentagon are [tex]$100^{\circ}, x^{\circ},(3 x+30)^{\circ}=(2 x-10)$[/tex]
(c) The sum of interior angle
4. The ratio of the interior to the exterior angles of a regular polygon [tex]$x: y$[/tex] and [tex]$80^{\circ}$[/tex]. Find [tex]$x^{\circ}$[/tex].
(a) Find the number of sides of the polygon in term of
(b) Find the number of sides of the polygon if [tex]$x: y=3: 2$[/tex]
Answer (2)
Calculate the exterior angle of the 7-sided polygon: Exterior Angle = 7 36 0 ∘ ≈ 51.4 3 ∘ .
Calculate the interior angle of the 7-sided polygon: Interior Angle = 18 0 ∘ − 51.4 3 ∘ ≈ 128.5 7 ∘ .
Determine the sum of the interior angles of the pentagon: ( 5 − 2 ) × 18 0 ∘ = 54 0 ∘ , and solve for x : x ≈ 56.6 7 ∘ .
Find the number of sides of the polygon in terms of x and y : n = y 2 x + 2 y , and when x : y = 3 : 2 , n = 5 .
n = 5
Explanation
Problem Overview and Strategy Let's break down this geometry problem step-by-step to make sure we understand each part clearly. We'll start with the polygons and their angles, and then move on to finding unknown values.
Understanding Regular Polygons A regular polygon has all its sides and angles equal. For a 7-sided regular polygon (a heptagon), we can find the exterior and interior angles using specific formulas.
Calculating the Exterior Angle The exterior angle of a regular polygon is found by dividing 360 degrees by the number of sides. In this case, the polygon has 7 sides. So, the exterior angle is: Exterior Angle = 7 36 0 ∘ ≈ 51.4 3 ∘ So, each exterior angle is approximately 51.4 3 ∘ .
Calculating the Interior Angle The interior angle of a regular polygon can be found by subtracting the exterior angle from 180 degrees. So, the interior angle is: Interior Angle = 18 0 ∘ − 51.4 3 ∘ ≈ 128.5 7 ∘ Therefore, each interior angle is approximately 128.5 7 ∘ .
Finding the Sum of Interior Angles of a Pentagon The sum of the interior angles of a pentagon (a 5-sided polygon) can be found using the formula ( n − 2 ) × 18 0 ∘ , where n is the number of sides. For a pentagon, n = 5 . So, the sum of the interior angles is: ( 5 − 2 ) × 18 0 ∘ = 3 × 18 0 ∘ = 54 0 ∘ Thus, the sum of the interior angles of any pentagon is 54 0 ∘ .
Solving for x in the Pentagon We are given the interior angles of a pentagon as 10 0 ∘ , x ∘ , ( 3 x + 30 ) ∘ , ( 2 x − 10 ) ∘ , and 8 0 ∘ . The sum of these angles must equal 54 0 ∘ . Therefore, we can write the equation: 100 + x + ( 3 x + 30 ) + ( 2 x − 10 ) + 80 = 540 Combine like terms: 6 x + 200 = 540 Subtract 200 from both sides: 6 x = 340 Divide by 6: x = 6 340 = 3 170 ≈ 56.6 7 ∘ So, the value of x is approximately 56.6 7 ∘ .
Finding the Number of Sides in Terms of x and y The ratio of the interior to the exterior angles of a regular polygon is given as x : y . We know that the interior angle is n 180 ( n − 2 ) and the exterior angle is n 360 , where n is the number of sides. Thus, we have: n 360 n 180 ( n − 2 ) = y x Simplify the equation: 360 180 ( n − 2 ) = y x 2 n − 2 = y x Solve for n :
n − 2 = y 2 x n = y 2 x + 2 n = y 2 x + 2 y So, the number of sides n in terms of x and y is y 2 x + 2 y .
Finding the Number of Sides When x:y = 3:2 If the ratio x : y = 3 : 2 , we can substitute these values into the equation for n :
n = 2 2 ( 3 ) + 2 ( 2 ) = 2 6 + 4 = 2 10 = 5 Therefore, the number of sides of the polygon is 5.
Final Answers
(a) The exterior angle of the 7-sided polygon is approximately 51.4 3 ∘ .
(b) The interior angle of the 7-sided polygon is approximately 128.5 7 ∘ .
(c) The sum of the interior angles of the pentagon is 54 0 ∘ , and x ≈ 56.6 7 ∘ .
(a) The number of sides of the polygon in terms of x and y is n = y 2 x + 2 y .
(b) If x : y = 3 : 2 , the number of sides of the polygon is 5.
Examples
Understanding polygons and their angles is crucial in various real-world applications. For instance, architects use these principles to design buildings with specific aesthetic and structural properties. Knowing the angles and side lengths helps in creating stable and visually appealing structures. Similarly, in computer graphics and game development, polygons are fundamental building blocks for creating 3D models and environments. The properties of polygons, such as the sum of interior angles, are used to ensure that the models are rendered correctly and efficiently. In everyday life, understanding these concepts can help in tasks such as tiling floors or designing gardens, where the angles and shapes of the tiles or garden beds need to fit together perfectly.
The exterior angle of a regular 7-sided polygon is approximately 51.43 degrees, and the interior angle is about 128.57 degrees. The sum of the interior angles of a pentagon is 540 degrees, leading to a value of x around 56.67 degrees. Additionally, the polygon with a ratio of interior to exterior angles of 3:2 has 5 sides.
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