ENGINEERING DRAWING ORTHOGRAPHIC PROJECTIONS

Area of a Right-Angled Triangle

Consider a rectangle of length l cm and width w cm. Draw a diagonal and cut out the rectangle.  Then cut along the diagonal to form two right-angled triangles. By arranging one triangle over the other, we find that the triangles are congruent.  In other words, the triangles are the same size and thus, equal in area.  This suggests that the area of a triangle is equal to half the area of a rectangle around it.  Therefore: In the diagram, we notice that the length of the rectangle is one side of the triangle.  This is said to be the base of the triangle.  So: Base of the triangle = Length of the rectangle The distance from the top of the triangle to the base is called the height of the triangle.  Therefore: Height of the triangle = Width of the rectangle Replacing l and w with the Base and Height in equation (1), we obtain: Using the pronumerals A for area, b for base and h for height, we can write the formula for the area of a right-angled triangle as: Area of a Triangle Consider the following triangle. Enclose the triangle by drawing a rectangle around it as shown below. It is clear from the diagram that the length of the rectangle is one side of the triangle.  This is said to be the base of the triangle.  So: Base of the triangle = Length of the rectangle The distance from the top of the triangle to the base is called the height of the triangle.  Clearly: Height of the triangle = Width of the rectangle Using the pronumerals A for area, b for base and h for height, we can write the formula for the area of a triangle as: Note: The rule (or equation) represents the relationship between the base and height of a triangle and its area.  Such an equation, which gives a rule for working out the value of one quantity from the values of others is called a formula. Just to recap the ongoing discussion:
A triangle with base b units and height h units has an area of A square units given by the formula
Example Find the area of a triangle with base 8 cm and height 5 cm. Solution

ENGINEERING DRAWING ORTHOGRAPHIC PROJECTIONS

Polygons

A polygon is a closed plane figure with three or more sides that are all straight. Some examples of polygons are shown below. The following figure is not a polygon as it is not a closed figure. A circle is not a polygon as it does not have straight sides. Polygons are named according to the number of sides.  The names of the most common polygons are given below: Concave Polygon If a polygon has a reflex angle, then it is said to be a concave polygon. An example of concave polygon is shown below. Convex Polygon If a polygon has no reflex angle, then it is said to be a convex polygon. Examples of the convex polygons are shown below. Regular Polygon A regular polygon's sides are all of the same length and its angles are the same size. For example, a square is a regular polygon. Examples of regular polygons are shown below. Irregular Polygon If a polygon is not a regular polygon, then it is said to be an irregular polygon. For example, the quadrilateral shown below is an irregular polygon.

ENGINEERING DRAWING ORTHOGRAPHIC PROJECTIONS

Quadrilaterals

A quadrilateral is a 2-dimensional closed shape with four straight sides.  E.g. The shape ABCD shown here is a quadrilateral. A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal of the quadrilateral.  AC is a diagonal of quadrilateral ABCD, as is BD. Types of Quadrilaterals There are seven types of quadrilaterals that can be divided into two groups:  parallelograms and other quadrilaterals. Parallelograms Quadrilaterals are called parallelograms if both pairs of opposite sides are equal and parallel to each other.  Different parallelograms and their properties are described below. Parallelogram Opposite sides of a parallelogram are parallel and equal in length. Opposite angles are equal in size. Note: Rectangle Opposite sides of a rectangle are parallel and equal in length. All angles are equal to 90°. Square Opposite sides of a square are parallel and all sides are equal in length. All angles are equal to 90°. Rhombus All sides of a rhombus are equal in length Opposite sides are parallel. Opposite angles of a rhombus are equal. The diagonals of a rhombus bisect each other at right angles. Note: Rectangles, squares and rhombuses (or diamonds) are parallelograms. Other Quadrilaterals Other quadrilaterals include trapeziums, kites and irregular quadrilaterals.

ENGINEERING DRAWING ORTHOGRAPHIC PROJECTIONS

Triangles

A plane is a flat or level surface in two dimensions.  Figures such as circles or squares have all of their parts lying on a plane and thus, are examples of plane figures. A triangle is a closed plane figure bounded by three line segments. Types of Triangles Triangles can be classified according to the length of their sides and the size of their angles. Classifying Triangles according to the Length of their Sides Scalene Triangles A scalene triangle has no equal sides. Equilateral Triangles An equilateral triangle has all sides equal. All angles of an equilateral triangle are 60°. Note: To show that the sides of an equilateral triangle have the same length, we place identical marks on the sides of the triangle. To show that the angles of an equilateral triangle are equal in size, we place identical curves on each angle. Isosceles Triangles An isosceles triangle has two sides equal. The angles opposite the equal sides are equal. Note: In the diagram, side BC has a different length to AB and AC. The side of the isosceles triangle that has a different length is said to be the base of the triangle.  So, BC is the base. Identical marks on the sides indicate that the sides are equal. Identical marks on the angles indicate that the angles are equal. Example 1 Classify each of the following triangles according to the lengths of their sides: Solution: Classifying Triangles according to the Size of their Angles Acute-angled Triangles An acute-angled triangle has all angles less than 90º (i.e. all three angles are acute). Obtuse-angled Triangles An obtuse-angled triangle has one angle greater than 90º.  That is, one angle is obtuse. Right-angled Triangles A right-angled triangle has one angle equal to 90°.  That is, one angle is a right angle. Note: The side opposite the right angle is called the hypotenuse. The hypotenuse is the longest side of the triangle (which can be verified with a ruler). Example 2 Classify each of the following triangles according to the size of their angles Solution: Activity 1