Contents:
Congruence: Two figures are congruent if they have the same shape and size. Two triangles are congruent if:
(i) the sides of one are equal to the sides of the other,
(ii) one side and two angles of one are equal to that of the other,
(iii) two sides and the angle between them of one are equal to the two sides and the angle between them of the other,
(iv) both triangles are right angled with the hypotenuse and another side equal to that of the other.
Example:
Example:If a square, rhombus, rectangle, or parallelogram are divided corner to corner they each produce congruent triangles.
Example: ABC is an equilateral triangle XYZ are points on it's sides where AX = BY = CZ. Prove that XYZ is also an equilateral triangle.
The angles A, B, C, are equal in a equilateral triangle. The sides are also equal therefore if AX = BY = CZ, then so does AZ = BX = CY.
If triangles have two sides and the angle between them equal then the triangles are congruent. This results in the lengths XZ = ZY = YX all being equal length.
If a triangle has all of its sides of equal length it is equilateral.
Example: X is the midpoint on the side BC, if AX is extended to double its original length to Y Show that ABYC is a parallelogram.
A paralellogram has two sets of parallel sides. It can be built up with two sets of congruent triangles.
(i) ABX = CYX because BX = CX, AX = YX, and AB = YC.
(ii) AXC = BXY because AX = YX, CX = BX, and AC = BY.
Similarity: Two figures are similar if they have the same shape even if they are of different size. Two triangles are similar if:
(i) The angles of one triangle are equal to the angles of the other.
(ii) The ratio of two sides are equal and the enclosed angle equal.
(iii) The ratio of all the sides are equal.
Example:
Enlargement: If the scale factor is greater than one the shape becomes bigger. If the scale factor is less than one the shape becomes smaller. If the scale factor is negative the shape goes through a rotation of 180
.
Shape A is enlarged by a scale of -2.
Therefore B is enlarged by a scale of -
.
The relative distance of old points and new points from the Centre of Enlargement are in the same scale, ie. x to y =1, x to z = 2.
The areas and volumes associated with enlargements are greater than the scale factor. For a scale factor 2, the lengths are twice as big, the areas are four times as big and the volumes are eight times as big.
The Rule: Sides are n times greater;
The Areas are n
times greater;
The Volumes are n
times greater.
Ratios: Scale factor: 5 : 2 Lengths: 5 : 2 Areas: 25 : 4 Volumes: 125 : 8
Top of Page
Top of Congruence
©Mathstutor.com 2001-2005
