can anyone help me with this! :)

Question

a). Calculate the exact perimeter of the triangle shown on the grid below. show work ! :)

 b. ) Then, the area of any triangle can be found without using the traditional formula involving the base and height. The formula is , where s is ½ the perimeter of the triangle and a, b , and c are the lengths of the sides of the triangle. Show how to use this formula to determine the area of the triangle shown above on the grid. You may give decimal answer, but no rounding should be done until you round the final answer.

Answer 1

a)

The exact perimeter of the triangle P = ?

From the above figure We need to find perimeter of triangle ABC.

The exact perimeter of the triangle P = AB + BC + AC

Coordinates of the triangle are A (0 , 4) B (-4 , 0) and C (4 , -4)

Length between points  A (0 , 4) and B (-4 , 0) = AB

AB = √ [(-4-0)²+(0-4)²]

AB = √ [16+16]

AB = √ [32]

AB = √ [2*16]

AB = 4√2

Length between points  A (0 , 4) and C (4 , -4) = AC

AC = √ [(4-0)²+(-4-4)²]

AC = √ [16+64]

AC = √ [80]

AC = √ [5*16]

AC = 4√5

Length between points  B (-4 , 0) and C (4 , -4) = BC

BC = √ [(4+4)²+(-4-0)²]

BC = √ [64+16]

BC = √ [80]

BC = √ [5*16]

BC = 4√5

The exact perimeter of the triangle P = AB + BC + AC

Substitute values : AB = 4√2 , AC = 4√5 and  BC = 4√5

P = 4√2 + 4√5 + 4√5

P = 4(√2 + 2√5).

The exact perimeter of the triangle is 4(√2 + 2√5).

Comments

how can i write the answer without the "( , )" please and thank you

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We can solve further

P = 4(√2 + 2√5)

P = 4√2 + 4*2√5

P = 4√2 + 8√5

Substitute : √2 = 1.4142 √5 = 2.2361

P = 4*1.4142  + 8*2.2361

P = 5.6568 + 17.8888

P = 23.5456.

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is it ^4sqr2+2sqr5 ?

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No,

4 (sqrt2 + 2 sqrt5 ) = 4 sqrt 2 + 8 sqrt 5

Answer 2

b)

The area of the triangle A = ?

From the above figure We need to find perimeter of triangle ABC.

Sides of triangle a = BC , b = AC , c = AB.

The exact perimeter of the triangle P = AB + BC + AC

Coordinates of the triangle are A (0 , 4) B (-4 , 0) and C (4 , -4)

Length between points  B (-4 , 0) and C (4 , -4) = BC = a

BC = √ [(4+4)²+(-4-0)²]

BC = √ [64+16]

BC = √ [80]

BC = √ [5*16]

a = BC = 4√5

Length between points  A (0 , 4) and C (4 , -4) = AC = b

AC = √ [(4-0)²+(-4-4)²]

AC = √ [16+64]

AC = √ [80]

AC = √ [5*16]

b = AC = 4√5

Length between points  A (0 , 4) and B (-4 , 0) = AB = c

AB = √ [(-4-0)²+(0-4)²]

AB = √ [16+16]

AB = √ [32]

AB = √ [2*16]

c = AB = 4√2

The exact perimeter of the triangle P = a + b + c

Substitute values : c = 4√2 , b = 4√5 and  a = 4√5

P = 4√5 + 4√5 + 4√2

P = 4(√2 + 2√5).

s = P/2

s = 4(√2 + 2√5)/2

s = 2(√2 + 2√5)

s = 2√2 + 4√5

s-a = 2√2 + 4√5 - 4√5

s-a = 2√2

s-b = 2√2 + 4√5 - 4√5

s-b = 2√2

s-c = 2√2 + 4√5 - 4√2

s-c = 4√5 - 2√2

s(s-a)(s-b)(s-c) = (2√2 + 4√5)(2√2)(2√2)(4√5 - 2√2)

= 8(2√2 + 4√5)(4√5 - 2√2)

= 8(8√10 - 4√4 + 16√25 - 8√10)

= 8(16*5 - 8)

= 8(80 - 8)

= 8(72)

= 576

The area of the triangle A = √[s(s-a)(s-b)(s-c)]

A = √576

A = 24

The Area of the triangle is 24.