This pdf includes the following topics:- Trapezium Definition Trapezium Basic Concept Properties of a Trapezium Area of a Trapezium
1. Week 1 & 2 Trapezum Area of trapezium is the region covered by a trapezium in a two- dimensional plane. It is the space enclosed in 2D geometry. A trapezium is a 2D shape which falls under category of quadrilaterals. Similar to other geometrical shapes, it also has its own properties and formulas based on area and perimeter. Let us learn in detail. Trapezium Definition A trapezium is a quadrilateral, which is defined as a shape with four sides and one set of parallel sides. Apart from trapezium, there are four more types of quadrilaterals. They are: 1 Parallelogram 2 Rectangle 3 Square 4 Rhombus All these quadrilaterals have one common property, which is, the sum of all the angles is 3600. Trapezium Basic Concept 1 The pair of parallel sides are called the bases while the non- parallel sides are called the legs of the trapezoid 2 The line segment connecting the midpoints of the non-parallel sides of a trapezoid is called the mid-segment 3 Check above the different types of trapezium figures, where arrow represents the parallel side of it. In all the three figures you can see, the two sides are parallel to each other, whereas the other two sides are non-parallel
2. 4 If we draw a line segment, between the two non-parallel sides, from the mid-point of both sides, the trapezium will be divided into two unequal parts. 5 You must have learned of isosceles triangles, where the two sides of a triangle are equal and the angle opposite the equal sides are also equal. In the same way, we have a figure, which is stated as Isosceles Trapezium, where the two non- parallel sides are equal and form equal angles at one of the bases. You can see the example of it, in the third figure given above. Properties of a Trapezium Here, we are going to learn about some more properties of the trapezium, which is also called as a trapezoid. A trapezium has the following properties: 1 Like other quadrilaterals, the sum of all the four angles of the trapezium is equal to 3600 2 A Trapezium has 4 unequal sides 3 A Trapezium has two parallel sides and two non-parallel sides 4 The diagonals of trapezium bisect each other 5 The length of the mid-segment is equal to 1/2 the sum of the parallel bases, in a trapezium 6 Two pairs of adjacent angles of a trapezium add up to 180 degrees Area of a Trapezium Trapezium area can be calculated by using the below formula: • Area = (1/2) h (AB+CD)
3. Perimeter of Trapezium The perimeter of trapezium formula is given by: • Perimeter = Sum of all the sides = AB + BC + CD + DA Derivation of Area of a Trapezium Following is the derivation for computing the area of the trapezium: The area of a trapezoid is equal to the sum of the areas of the two triangles and the area of the rectangle. We know that area of trapezoid = area of triangle 1 + area of rectangle + area of triangle 2. That means, A = (ah/2) + b1h + (ch/2) A = (ah + 2b1h + ch)/2 Simplifying the equation, rearranging the terms, and factoring result to: A = h/2[b1 + (a + b1 + h)] ….(i) If we assume the longer base of the trapezoid be b2, then b2 = a + b1 + h …..(ii) Substituting (ii) in equation (i), A = h/2(b1 + b2) Therefore the area of a trapezoid with bases b1, b2 and altitude h A = h/2(b1+b2) Applications of Trapezium The concept is a highly used concept in various physics computations and other mathematical calculations. This is the basis for obtaining the equations of motion as described in the 9th CBSE science textbook. The blend of the physics equations and mathematical calculations is very well explained to clear the level of understanding of a budding engineering mind. For the complete understanding and concept clearance from the BYJU’S one must go through the practice worksheets to check their hands on the concept and test learning. Frequently Asked Questions on Area of
4. What is the formula for Area of a trapezium? The formula to calculate area of trapezium is: Area = ½ x Sum of parallel sides x Distance between the parallel What are the formulas for Area and Perimeter of Area of trapezium = ½ x (a+b) x h Perimeter of trapezium = a+b+c+d Where a, b, c and d are the length of sides of a trapezium And h is the distance between the two parallel sides i.e a and b. What are the properties of Trapezium? A trapezium has 4 unequal sides: two parallel and two non-parallel Sum of all interior angles is 360 degrees Diagonals bisect each other How do you find the area of Trapezium? The area of trapezium depends on its parallel sides and distance between the parallel sides. If we know the length of parallel sides and the distance between them, then we can easily find the area of trapezium. A parallelogram is a two-dimensional geometrical shape, whose sides are parallel with each other. It is made up of four sides, where the pair of parallel sides are equal in length. Also, the opposite angles of a parallelogram are equal to each other. The area of parallelogram depends on the base and height of it. In geometry, you must have learned about many 2D shapes and sizes such as circle, square, rectangle, rhombus, etc. All of these shapes have a different set of properties. Also, the area and perimeter formulas of these shapes vary with each other, used to solve many problems. Let us learn here the definition, formulas and properties of a parallelogram. Table of contents: • Definition • Formula • Properties • Types • Theorems • Examples
5. Parallelogram Definition A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length and the opposite angles are equal in measure. In the figure above, you can see, ABCD is a parallelogram, where AB//CD and AD//BC. Also, AB = CD and AD = BC And, ∠A = ∠C & ∠B = ∠D Also, read: 5 Diagonal of a Parallelogram Formula 6 Important Questions Class 9 Maths Chapter 9 Areas Parallelograms Parallelogram Formula The formula for area and parameter of a parallelogram covered here in this section. Students can use these formulas and solve problems based on them. Area of Parallelogram Area of a parallelogram is the region occupied by it in a two- dimensional plane. Below is the formula to find the parallelogram Area = Base × Height
6. In the above figure, //gramABCD, Area is given by; Area = a b sin A = b a sin B where a is the slant length of the side of //gramABCD and b is the Check here: Area of a Parallelogram Formula Perimeter of Parallelogram The perimeter of any shape is the total distance of the covered around the shape or its total length of any shape. Similarly, the perimeter of a parallelogram is the total distance of the boundaries of the parallelogram. To calculate the perimeter value we have to know the values of its length and breadth. The parallelogram has its opposite sides equal in length. Therefore, the formula of the perimeter could be written as; Perimeter = 2 (a+b) Where a and b are the length of the equal sides of the Properties of Parallelogram If a quadrilateral has a pair of parallel opposite sides, then it’s a special polygon called Parallelogram. The properties of a parallelogram are as follows: 6 The opposite sides are congruent. 7 The opposite angles are congruent. 8 The consecutive angles are supplementary. 9 If anyone of the angles is a right angle, then all the other angles will be right. 10 The two diagonals bisect each other. 11 Each diagonal bisects the parallelogram into two congruent triangles. 12 The diagonals separate it into congruent.
7. Types of Parallelogram There are mainly four types of Parallelogram depending on various factors. The factors which distinguish between all of these different types of parallelogram are angles, sides etc. 7 In a parallelogram, say PQRS If PQ = QR = RS = SP are the equal sides, then it’s a rhombus. All the properties are the same for rhombus as for parallelogram. 8 Other two special types of a parallelogram are: • Rectangle • Square Is Square a Parallelogram? Square could be considered as a parallelogram since the opposite sides are parallel to each other and the diagonals of the square bisect each other. Is Rectangle a Parallelogram? Yes, a rectangle is also a parallelogram, because satisfies the conditions or meet the properties of parallelogram such as the opposite sides are parallel and diagonals intersect at 90 degrees. Parallelogram Theorems Theorem 1: Parallelograms on the same base and between the same parallel sides are equal in area. Proof: Two parallelograms ABCD and ABEF, on the same base DC and between the same parallel line AB and FC. To prove that area (ABCD) = area (ABEF). Parallelogram ABCD and rectangle ABML are on the same base and between the same parallels AB and LC. area of parallelogram ABCD = area of parallelogram ABML We know that area of a rectangle = length x breadth Therefore, area of parallelogram ABCD = AB x AL Hence, the area of a parallelogram is the product of any base of it and the corresponding altitude.
8. In ∆ADF and ∆BCE, AD=BC (∴ABCD is a parallelogram ∴ AD=BC) AF=BE (∴ABEF is a parallelogram ∴AF=BE) ∠ADF=∠BCE (Corresponding Angles) ∠AFD=∠BEC (Corresponding Angles) ∠DAF =∠CBE (Angle Sum Property) ∆ADE ≅ ∆BCF (From SAS-rule) Area(ADF) = Area(BCE) (By congruence area axiom) Area(ABCD)=Area(ABED) + Area(BCE) Hence, the area of parallelograms on the same base and between the same parallel sides is equal. Corollary: A parallelogram and a rectangle on the same base and between the same parallels are equal in area. Proof: Since a rectangle is also a parallelogram so, the result is a direct consequence of the above theorem. Theorem: The area of a parallelogram is the product of its base and the corresponding altitude. Given: In a parallelogram ABCD, AB is the base. To prove that Area(||gmABCD) = AB×AL Construction: Complete the rectangle ALMB by Drawing BM
9. perpendicular to CD. Examples of Parallelogram Example- Find the area of a parallelogram whose base is 5 cm and height is 8 cm. Solution- Given, Base = 5 cm and Height = 8 cm. We know, Area = Base x Height Area = 5 × 8 Area = 40 Sq.cm Example: Find the area of a parallelogram having length of diagonals to be 10 and 22 cm and an intersecting angle to be 65 degrees. Solution: We know that the diagonals of a parallelogram bisect each other, hence the length of half the diagonal will be 5 and 11 cm. The angle opposite to the side b comes out to be 180 – 65 = 115° We use the law of cosines to calculate the base of the parallelogram – b² = 5² + 11² – 2(11)(5)cos(115°) b² = 25 + 121 – 110(-.422) b² = 192.48 b = 13.87 cm.
10. After finding the base we need to calculate the height of the given To find the height we have to calculate the value of θ, so we use sine law 5/sin(θ) = b/sin(115) θ = 19.06 Now we extend the base and draw in the height of the figure and denote it as The right-angled triangle (marked with red line) has the Hypotenuse to be 22 cm and Perpendicular to be h. sin θ = h/22 h = 7.184 cm Area = base × height A = 13.87 × 7.184 A = 99.645 sq.cm
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