Find the missing length QG. Round answer to nearest tenth.

The answer to your question is QG = 55.8 ft

Step-by-step explanation:

Data

QG = x

Process

1.- Find T

The sum of the internal angles in a triangle equals 180°

          G  +  Q  +  T  = 180

-Solve for T

            T  =  180 - G  -  Q

- Substitution

            T = 180 - 41 - 67

-Simplification

             T = 72°

2.- Use the law of sines to find QG

              QG / sin T = GT / sin Q

-Solve for QG

              QG = GT sinT / sin Q

-Substitution

              QG = 54 sin 72 / sin 67

- Simplification

              QG = 54 (0.951)/0.921

-Result

             QG = 55.8 ft        

Step-by-step explanation:

Considering the given triangle QGT, to determine QG, we would apply the sine rule. It is expressed as

a/SinA = b/SinB = c/SinC

Where a, b and c are the length of each side of the triangle and angle A, Angle B and angle C are the corresponding angles of the triangle. Likening it to the given triangle, the expression becomes

QG/SinT = GT/SinQ = QT/SinG

The sum of the angles in a triangle is 180°. It means that

T = 180 - (41 + 67) = 72°

Therefore

QG/Sin72 = 54/Sin 67

Cross multiplying, it becomes

QGSin67 = 54Sin72

0.921QG = 54 × 0.951

0.921QG = 51.354

QG = 51.354/0.921

QG = 55.8 ft

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

what is the information provided?

step-by-step explanation: