Similarity:
Introduction to similarity:
Let us learn one of the most interesting topics,that of Similarity.We can see many things around us of the same shape and size,now let us learn about the concept of similarity in detail.
Two geometrical objects are called similar if they both have the same shape. More precisely, one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly "stretching" the same amount on all directions, possibly with additional rotation and reflection, i.e., both have the same shape, and one has the same shape as the mirror image of the other.
Study Axioms of Similarity:
We can understand the concept of similarity by studying about the different axioms as well.
AAA axiom of similarity: If in any two triangles, corresponding angles are equal then the two triangles are said to be similar to each other.
SSS axiom of similarity: If in any two triangles, corresponding sides are in same proportion then the two triangles are similar.
SAS axiom of similarity: If in any two triangles, two sides are in same proportion and if the included angle is equal then the triangles are similar.
Special Note 1: In a triangle ABC if a parallel line is drawn to any of the sides( Suppose BC) touching other sides at D and E(AB and AC respectively), then triangle ABC will be similar to the triangle formed, i.e ADE.
Special Note 2: In a right triangle ABC right angled at B, if a perpendicular is drawn from B to the side AC to meet at D, then the triangles thus formed and the original triangle are similar to each other.
Study Properties of Similarity in Triangles:
When two triangles(trABC,tr DEF) are similar to each other
1) The corresponding sides are in same proportion to each other, i.e
AB/DE = BC/EF = CA/FD
2) The corresponding angles in these triangles are equal
angle A = Angle D; angle B = angle E; angle C = angle F
3) The perimeters are in same proportion as the corresponding sides ratio.
perimeter(tr ABC) / perimeter(tr DEF) = AB/DE = BC/EF = CA/FD
4) The areas of the triangles are in same proportion as the square of ratios of sides
area(ABC) / area(DEF) = (AB/DE)^2 = (BC/EF)^2 = (CA/FD)^2.
Hope you like the above example of Similarity.Please leave your comments, if you have any doubts.
Introduction to similarity:
Let us learn one of the most interesting topics,that of Similarity.We can see many things around us of the same shape and size,now let us learn about the concept of similarity in detail.
Two geometrical objects are called similar if they both have the same shape. More precisely, one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly "stretching" the same amount on all directions, possibly with additional rotation and reflection, i.e., both have the same shape, and one has the same shape as the mirror image of the other.
Study Axioms of Similarity:
We can understand the concept of similarity by studying about the different axioms as well.
AAA axiom of similarity: If in any two triangles, corresponding angles are equal then the two triangles are said to be similar to each other.
SSS axiom of similarity: If in any two triangles, corresponding sides are in same proportion then the two triangles are similar.
SAS axiom of similarity: If in any two triangles, two sides are in same proportion and if the included angle is equal then the triangles are similar.
Special Note 1: In a triangle ABC if a parallel line is drawn to any of the sides( Suppose BC) touching other sides at D and E(AB and AC respectively), then triangle ABC will be similar to the triangle formed, i.e ADE.
Special Note 2: In a right triangle ABC right angled at B, if a perpendicular is drawn from B to the side AC to meet at D, then the triangles thus formed and the original triangle are similar to each other.
Study Properties of Similarity in Triangles:
When two triangles(trABC,tr DEF) are similar to each other
1) The corresponding sides are in same proportion to each other, i.e
AB/DE = BC/EF = CA/FD
2) The corresponding angles in these triangles are equal
angle A = Angle D; angle B = angle E; angle C = angle F
3) The perimeters are in same proportion as the corresponding sides ratio.
perimeter(tr ABC) / perimeter(tr DEF) = AB/DE = BC/EF = CA/FD
4) The areas of the triangles are in same proportion as the square of ratios of sides
area(ABC) / area(DEF) = (AB/DE)^2 = (BC/EF)^2 = (CA/FD)^2.
Hope you like the above example of Similarity.Please leave your comments, if you have any doubts.
No comments:
Post a Comment