Question

A right triangle has the following vertices Find the area of the triangle

(7,-3) (4,-3) (4,9)
20 pnts

Answer 1

Area = 18 square units

Explanation:

To find the area of the triangle, let's go through the following steps:

(i) Let the vertices be;

A = (7, -3)

B = (4, -3)

C = (4, 9)

(ii) The sides of the triangle are therefore,

AB, BC and CA

(iii) Using the distance formula, calculate the lengths of AB, BC and CA

`AB = √((7-4)^2 + ( -3 - (-3))^2)`

`AB = √(3^2 + (0)^2)\\`

`AB = √(9)`

`AB = 3`

`BC = √((4-4)^2 + ( -3 - 9)^2)`

`BC = √(0^2 + (-12)^2)`

`BC = √(144)`

`BC = 12`

`CA = √((4-7)^2 + ( 9 - (-3))^2)`

`CA = √((-3)^2 + (12)^2)`

`CA = √(9 + 144)`

`CA = √(153)`

`CA = 12.4`

(iv) Now that we have all the sides, let's calculate the area of the triangle using the Heron's formula.

Area =
`√(p(p-a)(p-b)(p-c))`

Where;

p =
`(a + b + c)/(2)`

a, b and c are the sides of the triangle.

In our case,

let

a = AB = 3

b = BC = 12

c = CA = 12.4

∴ p =
`(3 + 12 + 12.4)/(2)`

p = 13.7

Area =
`√(p(p-a)(p-b)(p-c))`

Area =
`√(13.7(13.7-3)(13.7-12)(13.7-12.4))`

Area =
`√(13.7(10.7)(1.7)(1.3))`

Area =
`√(323.9639)`

Area = 17.999

Area = 18 square units

OR

To get the area of the triangle, we can use a much simpler approach.

Since the triangle is a right triangle,

(i) the hypotenuse, which is the longest side is CA = 12.4

(ii) the other two sides are AB and BC. These two sides will form the right angle.

Therefore, we can use the relation:

Area =
`(1)/(2)` x base x height

Where;

the base or height can either be AB or BC

Area =
`(1)/(2)` x 3 x 12

Area = 18 square units

PS: In a right triangle, the other two sides apart from the hypotenuse form the right angle.