Verified answer

(ℓ1) : x - y + 1 = 0

(ℓ2) : 2y + 4 = 0 → 2y = - 4 → y = - 2 ← this is an horizontal line

(ℓ3) : 6x + 2y + 3 = 0

Point A: intersection (ℓ1) & (ℓ2) → you start with (ℓ1)

x - y + 1 = 0

x = y - 1 → recall (ℓ2): y = - 2

x = - 2 - 1 = - 3

→ A (- 3 ; - 2)

Point B: intersection (ℓ3) & (ℓ2) → you start with (ℓ3)

6x + 2y + 3 = 0

6x = - 2y - 3 → recall (ℓ2): y = - 2

6x = 4 - 3 = 1

x = 1/6

→ B (1/6 ; - 2)

Point C: intersection (ℓ1) & (ℓ3)

(ℓ1) : x - y + 1 = 0 → y = x + 1

(ℓ3) : 6x + 2y + 3 = 0 → recall the previous result

6x + 2.(x + 1) + 3 = 0

6x + 2x + 2 + 3 = 0

8x = - 5

x = - 5/8

Recall: y = x + 1

y = - (5/8) + (8/8)

y = 3/8

→ C (- 5/8 ; 3/8)

Distance AB

xAB = xB - xA = (1/6) + 3 = 19/6

yAB = yB - yA = - 2 + 2 = 0

AB² = xAB² + yAB² = (19/6)² + 0 = (19/6)²

AB = 19/6 ← this is a

Distance BC

xBC = xC - xB = - (5/8) - (1/6) = - 19/24

yBC = yC - yB = (3/8) + 2 = 19/8

BC² = xBC² + yBC² = (- 19/24)² + (19/8)² = (19²/24²) + (19²/8²) = 3610/24²

BC = (√3610)/24 ← this is b

Distance AC

xAC = xC - xA = - (5/8) + 3 = 19/8

yAC = yC - yA = (3/8) + 2 = 19/8

AC² = xAC² + yAC² = (19/8)² + (19/8)² = 2.(19/8)²

AC = (19/8).√2 ← this is c

According the Heron's formula:

https://en.wikipedia.org/wiki/Heron%27s_formula

area = √[p.(p - a).(p - b).(p - c)] → where p is half of the perimeter

p = (a + b + c)/2

p = [(19/6) + (√3610)/24 + (19√2)/8]/2

p = [(76/24) + (√3610)/24 + (57√2)/24]/2

p = (76 + √3610 + 57√2)/48

p - a = [(76 + √3610 + 57√2)/48] - (19/6) = (- 76 + √3610 + 57√2)/48

p - b = [(76 + √3610 + 57√2)/48] - [(√3610)/24] = (76 - √3610 + 57√2)/48

p - c = [(76 + √3610 + 57√2)/48] - [(19/8).√2] = (76 + √3610 - 57√2)/48

= p.(p - a).(p - b).(p - c)

= [(76 + √3610 + 57√2)/48] * [(- 76 + √3610 + 57√2)/48] * [(76 - √3610 + 57√2)/48] * [(76 + √3610 - 57√2)/48]

= (1/48⁴).(76 + √3610 + 57√2).(- 76 + √3610 + 57√2).(76 - √3610 + 57√2).(76 + √3610 - 57√2)

= (1/48⁴).[76 + (√3610 + 57√2)].[- 76 + (√3610 + 57√2)].[76 - (√3610 - 57√2)].[76 + (√3610 - 57√2)]

= (1/48⁴).[- 76² + (√3610 + 57√2)²].[76² - (√3610 - 57√2)²]

= (1/48⁴).[- 5776 + (3610 + 114√7220 + 6498)].[5776 - (3610 - 114√7220 + 6498)]

= (1/48⁴).[- 5776 + 3610 + 114√7220 + 6498].[5776 - 3610 + 114√7220 - 6498]

= (1/48⁴).[4332 + 114√7220].[- 4332 + 114√7220]

= (1/48⁴).[- 4332² + (114² * 7220)]

= 75064896/48⁴

area = √[p.(p - a).(p - b).(p - c)]

area = √(75064896/48⁴)

area = (1/48²).√75064896

area = (1/48²).√(576 * 130321)

area = (1/48²).√(24² * 361²)

area = (24 * 361)/(48 * 48)

area = 361/96

Hola

L1) x - y + 1 = 0

L2) 2 y + 4 = 0

L3) 6 x + 2 y + 3 = 0

De L2)

y = -2

Intersección con L1)

x = y - 1 = -2 - 1 = -3

(-3 , -2)

Intersección con L3)

x = (-2y - 3)/6 = ((-2)(-2) - 3)/6 = (4-3)/6 = 1/6

( (1/6) , -2)

Base sobre L2

Base = (1/6) - (-3) = (1/6) + (18/6) = 19/6

Intersección de L1 ; L3

Componente y

x = y - 1 = (- 2 y - 3)/6

6 y - 6 = - 2 y - 3

8 y = 3

y = 3/8

Altura con respecto a linea horizontal L2

Altura = (3/8) - (-2) = (3/8) + (16/8) = 19/8

Superficie = (1/2) * (19/6) (19/8)

Superficie = 361/96

**************************

Saludos

Los vértices se encuentran en:

(-3, -2); (- 5/8, 3/8); (1/6, -2)

Con estos puntos, puedes determinar el área por determinantes.

-------------------

A = 361/96

A ≈ 3.76042

-------------------