Question

Solve for the area using heron´s formula

Answer 1

Given =>

A figure ABCD with two adjacent triangles i.e ABC &ACD

To Acquire =>

The Area of the figure

Points to know while solving this problem=>

• Area of a ∆ According to heron =
• [tex] \sqrt{s(s - a)(s - b)(s - c)} [/tex]
• Where, s= semi-perimeter
• a= length of side a
• b = length of side b
• с = length of side c
• Perimeter of a ∆ = sum of it's all 3 sides
• The area of the Given Figure would be Area of ∆ABC Area of ∆ACD

Step-by-step explanation:

In ∆ABC,

Perimeter of the ∆

= mAB +mBC+mAC

Where,

mAB =20cm

mAC = 16cm

mBC= ?

Also

mBC

= Base of the ∆ ABC where AB = perpendicular & AC = Hypotenuse

So

mBC = √AC²-AB².......(Using Pythagoras theorem)

mBC = √20²-16²

mBC =√400-256

mBC =√144

mBC = 12.....( The square root of 144 = 12)

Hence,

The perimeter of ∆

= mAB +mBC+mAC

= 16+12+20

= 48cm

Semiperimeter = 48/2

= > 24 cm

.°.

The area of ∆ ABC according to heron's formula

=>

$\sqrt{24(24 - 16)(24 - 12)(24 - 20)} \\ = > \sqrt{24 \times 8 \times 12 \times 4 } \\ = > \sqrt{12 \times 2 \times 8 \times 12 \times 4} \\ = > \sqrt{12 \times 12 \times 2 \times 4 \times 8} \\ = > \sqrt{12 \times 12 \times 8 \times 8} \\ = > 12 \times 8 \\ = > 96cm {}^{2}$

Now,

In ∆ACD,

The perimeter = mAC+mCD+mAD

where,

mAC=16cm

mCD=23cm

mAD=20cm

=>

$16cm + 23cm + 20cm \\ = > 59cm$

So...

Semiperimeter = 59/2cm

=> 29.5

Area of the Same∆ using Heron's formula

= 156 .97cm²

Hence,

The area of the figure

=>96cm²+156.97cm²

=>252.97 cm²

Answer 2

9514 1404 393

Answer:

  252.8 cm²

Step-by-step explanation:

The missing side of the right triangle can be found from the Pythagorean theorem:

  s² = 20² -16² = 400 -256 = 144

  s = 12 . . . . cm

The area of a right triangle is more easily found using the traditional area formula:

  A = 1/2bh

  A = 1/2(12 cm)(16 cm) = 96 cm² (left-side triangle)

The area of the triangle on the right can be found from Heron's formula. The semiperimeter is ...

  s = (16 +20 +23)/2 = 29.5

The area is ...

  A = √(29.5(29.5 -16)(29.5 -20)(29.5 -23)) = √(29.5·13.5·9.5·6.5)

  A = √24591.9375 ≈ 156.818 . . . . . cm² (right-side triangle)

Then the total area of the figure is ...

  A = 96 cm² +156.818 cm² = 252.818 cm² . . . . total area