69.4k views
2 votes
A right triangle has the following vertices Find the area of the triangle

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

1 Answer

6 votes

Answer:

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.

User Tony Wang
by
7.8k points

Related questions

asked Jul 7, 2024 234k views
Mikko asked Jul 7, 2024
by Mikko
7.6k points
1 answer
0 votes
234k views
asked Dec 12, 2021 59.4k views
Oriolpons asked Dec 12, 2021
by Oriolpons
7.4k points
2 answers
1 vote
59.4k views