three sides. The length from one vertex to another is found using - - PowerPoint PPT Presentation

DAY 116 – PERIMETER AND AREA

OF A TRIANGLE ON X-Y PLANE

INTRODUCTION

We have different ways in which we can use to find the area of a triangle. Some of these ways include using the formula

1 2 𝑐ℎ, 1 2 𝑏𝑐 sin 𝜄 and using

Heron’s formula. However, some of these methods may not help us to find the area of a triangle when we are only given the coordinates of its vertices. We have to perform a number of computations to use them. In this lesson, we will find the area and the perimeter of triangle on xy- plane.

VOCABULARY

Perimeter This is the distance around a shape on a plane.

Finding perimeter of a triangle on xy- plane From the definition of the perimeter, we find the perimeter of a triangle by adding the lengths of all the three sides. The length from one vertex to another is found using the distance formula. The distance formula If 𝐵(𝑦1, 𝑧1) and B(𝑦2, 𝑧2) are two points on xy- plane, the distance from A to B is given by the formula, (𝑦2 − 𝑦1)2+(𝑧2 − 𝑧1)2

Example 1 Find the length of a triangle with vertices and hence, the perimeter at 𝐵 −4, −4 , 𝐶(8, −4) and 𝐷(8,1). Solution Distance from A to B is 8 − −4 2 + (−4 − −4)2 = 12 Distance from B to C is −4 − 1 2 + (8 − 8)2 = 5 Distance from B to C is 8 − −4 2 + (1 − −4)2 = 13 Perimeter of ∆𝐵𝐶𝐷 = 𝐵𝐶 + 𝐶𝐷 + 𝐵𝐷 =12+5+13 = 30 𝑣𝑜𝑗𝑢𝑡

Finding the area of a triangle on xy- plane We can use Heron’s formula to find the area of a triangle after getting the perimeter. According to Heron’s formula, Area of Triangle = 𝑡(𝑡 − 𝑏)(𝑡 − 𝑐)(𝑡 − 𝑑) where s =

1 2 𝑢ℎ𝑓 𝑞𝑓𝑠𝑗𝑛𝑓𝑢𝑓𝑠 and a,b,c are lengths of

each side. If we use this formula to find area of the ∆𝐵𝐶𝐷 in example 1 above, 𝑡 =

30 2 , 𝑏 = 12, 𝑐 = 5 and 𝑑 = 13.

𝐵 = 15(15 − 12)(15 − 5)(15 − 13) 𝐵 = 15 × 3 × 10 × 2 𝐵 = 900 = 30 𝑡𝑟. 𝑣𝑜𝑗𝑢𝑡

However, Heron’s formula can be tedious especially when the lengths of the sides are irrational numbers. Given a triangle with vertices at 𝑦1, 𝑧1 , 𝑦2, 𝑧2 , (𝑦3, 𝑧3) a simpler way to find the area of triangle is to use the formula, 𝐵 =

1 2 (𝑦1 𝑧2 − 𝑧3 + 𝑦2 𝑧3 − 𝑧1 + 𝑦3 𝑧1 − 𝑧2 )

This formula can easily be remembered by finding half the value of the determinant of the matrix, 1 1 1 𝑦1 𝑦2 𝑦3 𝑧1 𝑧2 𝑧3

Example 2 Find the area of a triangle with vertices at 1,1 , 4,2 and (1,4). Solution Let 1,1 , 4,2 and (1,4) be 𝑦1, 𝑧1 , 𝑦2, 𝑧2 and (𝑦3, 𝑧3) 𝐵 =

1 2 (𝑦1 𝑧2 − 𝑧3 + 𝑦2 𝑧3 − 𝑧1 + 𝑦3 𝑧1 − 𝑧2 )

𝐵 =

1 2 (1 2 − 4 + 4 4 − 1 + 1(1 − 2)

𝐵 =

1 2 −2 + 4 3 − 1

𝐵 = 4.5 𝑡𝑟 𝑣𝑜𝑗𝑢𝑡

HOMEWORK Find the perimeter of a triangle with vertices at 4,2 , 12,2 𝑏𝑜𝑒 2,8 .

ANSWERS TO HOMEWORK

24 𝑣𝑜𝑗𝑢𝑡

THE END