{"id":1145,"date":"2020-10-15T14:36:34","date_gmt":"2020-10-15T14:36:34","guid":{"rendered":"http:\/\/themindpalace.in\/?p=1145"},"modified":"2021-08-26T05:32:50","modified_gmt":"2021-08-26T05:32:50","slug":"polygon","status":"publish","type":"post","link":"https:\/\/themindpalace.in\/index.php\/2020\/10\/15\/polygon\/","title":{"rendered":"Polygon"},"content":{"rendered":"\n

Summary of polygon<\/a><\/p>\n\n\n\n

Solved exercise polygon<\/a><\/p>\n\n\n\n

<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Summary<\/h1>\n\n\n\n

POLYGONS:A simple closed curve made up of only line segments is called a polygon. polygon comes from Greek word poly means “”many gon means \u00c4ngle”. Polygons are two dimensional shapes made up of straight lines and shapes is closed (all the lines connect up).<\/p>\n\n\n\n

Curves that are polygons<\/span>.<\/h4>\n\n\n\n
\"\"<\/figure><\/div>\n\n\n\n

Curves that are not polygons<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Classification of polygons<\/span><\/h4>\n\n\n\n
\"\"<\/figure><\/div>\n\n\n\n

Diagonals<\/span><\/strong><\/p>\n\n\n\n

A diagonal is a line segment connecting two non consecutive vertices of a polygon.<\/p>\n\n\n\n

Interior and exterior<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Simple and complex polygons<\/span><\/strong>.<\/h4>\n\n\n\n

A simple polygon has only one boundary and the sides do not cross each other wise it is a complex polygon.<\/p>\n\n\n\n

Simple polygon<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Complex polygon<\/span><\/strong><\/p>\n\n\n\n

Complex polygon is a polygon whose sides cross over each other one or more times<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Concave and convex polygons<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

A convex polygon has no internal angle more than 1800<\/sup> and there are any internal angles greater than a straight angle, then it is a concave polygon.<\/p>\n\n\n\n

Regular polygons<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

A regular polygon is both equiangular and equilateral.<\/p>\n\n\n\n

For Ex:  A square has sides of equal length and angles of equal measure. Hence it is a regular polygon.<\/p>\n\n\n\n

Irregular polygon.<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

A rectangle is equiangular but not equilateral. Hence it is not a regular polygon. It’s called an irregular polygon.<\/p>\n\n\n\n

Exercise 4.1<\/strong><\/h1>\n\n\n\n

1.Here are some given given figures classify each of them on the basis of their following.<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Ans:1)Simple curve <\/span><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

2)simple closed curve<\/span><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

3)Polygon<\/span><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

4)Convex polygon<\/span><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

5)Concave polygon.<\/span><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

2. How many diagonals does each of the following have?<\/span><\/p>\n\n\n\n

a) A convex quadrilateral<\/p>\n\n\n\n

 (i) ABCD is a convex quadrilateral which has two diagonals AC and BD.<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

(b) A regular hexagon<\/span><\/p>\n\n\n\n

ABCDEF is a regular hexagon which has nine diagonals AE, AD, AC, BF, BE, BD, CF, CE and DF.<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

(c) A triangle<\/p>\n\n\n\n

ABC is a triangle which has no diagonal<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

3. <\/span>What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and verify)<\/span><\/p>\n\n\n\n

Solution:In the given figure, we have a quadrilateral ABCD. Join AC diagonal which divides the quadrilateral into two triangles ABC and ADC.<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

In \u2206ABC, \u22203 + \u22204 + \u22206 = 180\u00b0\u2026(i) (angle sum property)<\/p>\n\n\n\n

In \u2206ADC, \u22201 + \u22202 + \u22205 = 180\u00b0 \u2026(ii) (angle sum property)<\/p>\n\n\n\n

Adding, (i) and (ii)<\/p>\n\n\n\n

\u22201 + \u22203 + \u22202 + \u22204 + \u22205 + \u22206 = 180\u00b0 + 180\u00b0<\/p>\n\n\n\n

\u21d2 \u2220A + \u2220C + \u2220D + \u2220B = 360\u00b0<\/p>\n\n\n\n

Hence, the sum of all the angles of a convex quadrilateral = 360\u00b0.<\/p>\n\n\n\n

A non-convex quadrilateral<\/p>\n\n\n\n

Yes, this property also holds true for a non-convex quadrilateral<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Examine the table. (Each figure is divided into triangles and the sum of the angles reduced from that).
<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

What can you say about the angle sum of a convex polygon with number of sides?
(a) 7
(b) 8
(c) 10
(d) n<\/p>\n\n\n\n

Solution:<\/span>
From the above table, we conclude that the sum of all the angles of a polygon of side \u2018n\u2019
= (n \u2013 2) \u00d7 180\u00b0<\/p>\n\n\n\n

(a) Number of sides = 7
Angles sum = (7 \u2013 2) \u00d7 180\u00b0 = 5 \u00d7 180\u00b0=9000<\/sup><\/p>\n\n\n\n

b) Number of sides = 8
Angle sum = (8 \u2013 2) \u00d7 180\u00b0 = 6 \u00d7 180\u00b0=10800<\/sup><\/p>\n\n\n\n

d) Number of sides = n
Angle sum = (n \u2013 2) \u00d7 180\u00b0<\/p>\n\n\n\n

5. What is a regular polygon? State the name of a regular polygon of
(i) 3 sides
(ii) 4 sides
(iii) 6 sides<\/p>\n\n\n\n

Solution:<\/span>
A polygon with equal sides and equal angles is called a regular polygon.<\/p>\n\n\n\n

(i) Equilateral triangle<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

(ii) Square<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

(iii) Regular Hexagon<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

6. Find the angle measure x in the following figures:<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Solution:<\/span>
(a) Angle sum of a quadrilateral = 360\u00b0<\/p>\n\n\n\n

\u21d2 50\u00b0 + 130\u00b0 + 120\u00b0 + x = 360\u00b0<\/p>\n\n\n\n

\u21d2 300\u00b0 + x = 360\u00b0<\/p>\n\n\n\n

\u21d2 x = 360\u00b0 \u2013 300\u00b0 = 60\u00b0<\/p>\n\n\n\n

X = 60\u00b0<\/strong><\/p>\n\n\n\n

(b) Angle sum of a pentagon = 540\u00b0<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

\u21d2 30\u00b0 + x + 110\u00b0 + 120\u00b0 + x = 540\u00b0 [\u2235 180\u00b0 \u2013 70\u00b0 = 110\u00b0; 180\u00b0 \u2013 60\u00b0 = 120\u00b0]<\/p>\n\n\n\n

\u21d2 2x + 260\u00b0 = 540\u00b0<\/p>\n\n\n\n

\u21d2 2x = 540\u00b0 \u2013 260\u00b0<\/p>\n\n\n\n

\u21d2 2x = 280\u00b0<\/p>\n\n\n\n

\u21d2 x = 140\u00b0<\/strong><\/p>\n\n\n\n

7. (a) Find x + y + z<\/span><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Solution:
(a) \u2220a + 30\u00b0 + 90\u00b0 = 180\u00b0 [Angle sum property]<\/p>\n\n\n\n

\u21d2 \u2220a + 120\u00b0 = 180\u00b0<\/p>\n\n\n\n

\u21d2 \u2220a = 180\u00b0 \u2013 120\u00b0 = 60\u00b0<\/p>\n\n\n\n

Now, y = 180\u00b0 \u2013 a (Linear pair)<\/p>\n\n\n\n

\u21d2 y = 180\u00b0 \u2013 60\u00b0<\/p>\n\n\n\n

\u21d2 y = 120\u00b0<\/p>\n\n\n\n

and, z + 30\u00b0 = 180\u00b0 [Linear pair]<\/p>\n\n\n\n

\u21d2 z = 180\u00b0 \u2013 30\u00b0 = 150\u00b0<\/p>\n\n\n\n

also, x + 90\u00b0 = 180\u00b0 [Linear pair]<\/p>\n\n\n\n

\u21d2 x = 180\u00b0 \u2013 90\u00b0 = 90\u00b0<\/p>\n\n\n\n

Thus x + y + z = 90\u00b0 + 120\u00b0 + 150\u00b0 = 360\u00b0<\/p>\n\n\n\n

Sum of the measures of the exterior angles of a polygon  = 360<\/strong>0<\/sup><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n
\"\"<\/figure><\/div>\n\n\n\n

Sum of the two angles in linear pair = 1800<\/sup><\/p>\n\n\n\n

Pentagon has 5 sides<\/p>\n\n\n\n

then sum of the interior angle of pentagon= (n-2) x1800<\/sup><\/p>\n\n\n\n

           = (5-2) x1800<\/sup><\/p>\n\n\n\n

   <\/sup>      = 3 x1800<\/sup>    = 5400<\/sup><\/p>\n\n\n\n

measure of each angle = 540\/5 <\/sup>      = 1080<\/sup><\/p>\n\n\n\n

then sum of the linear <\/sup>pair  = 1800 <\/sup>\u2013 108 = 720<\/sup><\/p>\n\n\n\n

we know that sum of all exterior angles of any polygon is 360<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n
\"\"<\/figure><\/div>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure><\/div>\n\n\n\n

Measure of each exterior angle of a regular polygon = 360\/n<\/p>\n\n\n\n

Find the measure of each exterior angle of a regular polygon of<\/span><\/p>\n\n\n\n

i)9 sides<\/p>\n\n\n\n

ii)15 sides<\/p>\n\n\n\n

Ans:1)Sum of all exterior angles of the given polygon =3600          <\/sup><\/p>\n\n\n\n

Each exterior angle of a regular polygon has the same measure.<\/p>\n\n\n\n

Thus measure of each exterior angle of a regular polygon of 9 sides<\/p>\n\n\n\n

360\/9=40<\/p>\n\n\n\n

3. How many sides does a regular polygon have if the measure of an exterior angle is 240<\/sup>?       <\/p>\n\n\n\n

Sum of all exterior angles of the given polygon is 3600<\/sup>\/24=15<\/p>\n\n\n\n

4. How many sides does a regular polygon have if each its interior angles is 1650<\/sup><\/span><\/p>\n\n\n\n

1800 <\/sup>Regular polygon = all the sides and angles are equal<\/p>\n\n\n\n

Let number of sides = n = measure of angles<\/p>\n\n\n\n

Sum of interior angles of a regular polygon = (n- 2)x1800<\/sup><\/p>\n\n\n\n

165 n = (n- 2)x1800<\/sup><\/p>\n\n\n\n

165 n = 1800<\/sup> n \u2013 360<\/p>\n\n\n\n

165 n – 1800<\/sup> n = \u2013 360<\/p>\n\n\n\n

– <\/sup>15 n = \u2013 360<\/p>\n\n\n\n

 <\/sup> n = 24 <\/sup><\/p>\n\n\n\n

165 n = (n- 2)x1800<\/sup><\/p>\n\n\n\n

165 n = 1800<\/sup> n \u2013 360<\/p>\n\n\n\n

165 n – 1800<\/sup> n = \u2013 360<\/p>\n\n\n\n

– <\/sup>15 n = \u2013 360<\/p>\n\n\n\n

 <\/sup> n = 24 <\/sup><\/p>\n\n\n\n

5.a)is it possible to have a regular polygon with measure of each exterior angle is<\/span><\/p>\n\n\n\n

b)Can it be a regular interior angle of regular polygon?why?<\/span><\/p>\n\n\n\n

Ans: The sum of all exterior angle of all polygon is 360.Also in a regular polygon each exterior angle is of the same measure. Hence if 360 is a perfect multiple of the given exterior angle then the given polygon will be possible.<\/p>\n\n\n\n

a)Exterior angle =220<\/sup>                     <\/p>\n\n\n\n

3600 is not a perfect multiple of 220<\/sup>. Hence such polygon is not possible.                              <\/p>\n\n\n\n

b)Interior angle =220<\/p>\n\n\n\n

Exterior angle = 1800<\/sup>-220<\/sup>=1580<\/sup><\/p>\n\n\n\n

Such a polygon is not possible as 3600<\/sup> is not a perfect multiple of 1580<\/sup><\/p>\n\n\n\n

6. What is the minimum interior angle possible for a regular polygon?<\/span><\/p>\n\n\n\n

b)what is the maximum exterior angle possible for a regular polygon?<\/span><\/p>\n\n\n\n

Consider a regular polygon having the lowest possible number of sides(i,e an equilateral triangle).The exterior angle of tthis triangle will be the maximum exterior angle possible for nay rtegular polygon.<\/p>\n\n\n\n

Exterior angle of an equilateral triangle=3600<\/sup>\/3<\/span>=1200<\/sup><\/p>\n\n\n\n

Hence,maximum possible measure of exterior angle for any polygon is        1200<\/sup><\/p>\n\n\n\n

we know that an exterior angle and an interior angle are always in a linear pair<\/p>\n\n\n\n

Hence, minimum interior angle=1800<\/sup>-1200<\/sup>=600     <\/sup><\/p>\n","protected":false},"excerpt":{"rendered":"

Summary of polygon Solved exercise polygon Summary POLYGONS:A simple closed curve made up of only line segments is called a polygon. polygon comes from Greek word poly means “”many gon means \u00c4ngle”. Polygons are...<\/p>\n","protected":false},"author":3,"featured_media":1148,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[475,14],"tags":[135,136],"class_list":["post-1145","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-posts-in-english","category-mathematics","tag-maths-polygons","tag-ncert-maths-polygons"],"cp_meta_data":{"_edit_lock":["1629955970:2"],"_edit_last":["2"],"_layout":["inherit"],"_oembed_95287caaddeb112cd4edfcbd8e525566":["