The perimeter and area of both a rhombus and a parallelogram can be - PowerPoint PPT Presentation

D AY 118 – P ERIMETER AND AREA OF A RHOMBUS AND A PARALLELOGRAM ON XY - PLANE

I NTRODUCTION The perimeter and area of rhombi and parallelograms have been dealt with in earlier lessons. Both the area and perimeter of these quadrilaterals can be easily determined by applying the appropriate formula. In coordinate geometry, it is possible to calculate their perimeter and area on the 𝑦𝑧 - plane by using coordinates of their vertices to calculate the required. In this lesson, we are going to learn how to find the perimeter and area of a rhombus and a parallelogram on the 𝑦𝑧 - plane.

V OCABULARY 1. Rhombus A quadrilateral whose sides are all congruent and whose diagonals are not congruent. 2. Parallelogram A quadrilateral with parallel opposite sides and opposite sides congruent.

The perimeter and area of both a rhombus and a parallelogram can be found by applying the formulae below if the dimensions are given: Perimeter of a rhombus = length of any side × 4 Perimeter a parallelogram = 2 length + width Area of a rhombus = length of any side × altitude Area of a parallelogram = length × altitude

We need to recall one important formula before we embark on the process of finding the perimeter and area of rhombi and parallelograms on the coordinate plane. This is called Heron’s formula . Consider ∆ABC below. A 𝑑 𝑐 B C 𝑏

According to Heron’s formula: 𝒃+𝒄+𝒅 𝒕 = 𝒕 The area of ∆ABC is given by: 𝒕 𝒕 − 𝒃 𝒕 − 𝒄 𝒕 − 𝒅

F INDING THE AREA AND PERIMETER OF A BOTH A RHOMBUS AND A PARALLELOGRAM ON THE X - Y PLANE The perimeter can be found in the same way as we used to find the perimeter of a square and a rectangle in our previous class. Since there exists a slight difference in the properties of a rhombus and a parallelogram, a similar procedure is used to find the area of both quadrilaterals.

Example The vertices of parallelogram JKLM are 𝐾 3,2 , 𝐶 8, 2 , 𝐷 10, 5 and 𝐸 5,5 . Calculate its perimeter and area. Solution We can sketch the parallelogram as shown below. M L J K

We need to calculate the length of any two adjacent sides. This will give us the dimensions if the parallelogram. Let us find the lengths of sides JK and KL. 8 − 3 2 + 2 − 2 2 = JK = 25 = 5 units 10 − 8 2 + 5 − 2 2 = KL = 13 units ∴ The perimeter = 2 5 + 13 = 17.21 units

In order to find the area, we need to divide the parallelogram into two congruent triangles and then find the area of each of the triangles using Heron’s formula. The area of the parallelogram will be the sum of the areas of the two congruent triangles. M L J K

Let us first get the length KM: 8 − 5 2 + 2 − 5 2 = KM = 18 units We use Heron’s formula to find the area of ∆JKM : 13 + 18 + 5 𝑡 = = 6.424 2 𝐵𝑠𝑓𝑏 = 𝑡 𝑡 − 𝑏 𝑡 − 𝑐 𝑡 − 𝑑

= 6.424 6.424 − 13 6.424 − 18 6.424 − 5 = 7.5 sq. units The parallelogram JKLM is therefore given as: 𝟖. 𝟔 × 𝟑 = 𝟐𝟔 𝐭𝐫. 𝐯𝐨𝐣𝐮𝐭 Note For the sake of accuracy the radical should be simplified at the last stage of the calculation.

HOMEWORK Find the perimeter of rhombus ABCD whose vertices are: 𝐵 2,2 , 𝐶 7, 2 , 𝐷 10, 6 and 𝐸 5,6 .

A NSWERS TO HOMEWORK Perimeter = 20 units

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