Question

40°

A
82⁰
C
Work out the area of triangle ABC
Give your answer to 1 decimal place
12 m
B
(5 marks)

Answer 1

94.1 m²

Explanation:

To find the area of the triangle.

Given:

In ∆ ABC

m ∠A= 82°

m ∠C = 40°

m ∠B =180° - 82° - 40° = 58°

AB = 12 m

Solution:

To find the area of the triangle, we can use the following Heron's formula:

`\sf Area = √(s(s-a)(s-b)(s-c))`

where

• a, b, and c are the lengths of the sides of the triangle.
• s is semi perimeter.

We are given the length of side AB (c) is 12 m. We also know the measures of angles B and C. To find the lengths of the other two sides, we can use the law of sines:

`\sf (a )/(sin(A) )= (b )/( sin(B) )= (c )/( sin(C))`

We can rewrite this formula as follows:

`\sf b = c \cdot ( sin(B) )/(sin(C))`

`\sf a = c \cdot ( sin(A) )/(sin(C))`

We know that the length of side AB(c) is 12 m, and the measures of angles A and C are 82° and 40°, respectively. We can use the above formulas to find the lengths of the other two sides:

`\sf AC(b) = 12 \cdot ( sin(58^\circ ) )/(sin(40^\circ )) \\\\ \approx 15.832`

`\sf BC( a) = 12 \cdot ( sin(82^\circ) )/(sin(40^\circ ))\\\\ \approx 18.487m`

Now that we know the lengths of all three sides of the triangle, we can find the semi perimeter:

`\begin{aligned} \sf Semi-Perimeter(s) &= \frac {AB + BC + AC }{2} \\\\& = (12 + 18.487 + 15.832 )/(2)\\\\ &= (46.319)/(2) \\\\ &= 23.159m \end{aligned}`

Now,

Finding area using Heron's formula:

`\begin{aligned}\sf Area & =√(23.159(23.159 - 12)(23.159 - 18.487)(23.159 - 15.832))\\\\ &= √( 23.159\cdot 11.159 \cdot 4.672\cdot 7.327 )\\\\ &= √(8846.553 )\\\\& = 94.056m^2 \end{aligned}`

Therefore, the area of the triangle is 94.1 m² in one decimal form.