Question

A surveyor took the following measurements from two irregularly shaped pieces of land. Some of the lengths and angle measures are missing. Find all missing lengths and angle measures. Round lengths to the nearest tenth and angle measures to the nearest minute

Answer 1

To find the width of the river, use trigonometry and the tangent function. The width of the river is approximately 67.1 meters.

Step-by-step explanation:

To find the width of the river, we can use trigonometry. The surveyor measured an angle of 35° from the baseline to the tree on the opposite bank. We know the distance along the river is 100 m. Let's denote the width of the river as 'x'.

We can use the tangent function to solve for 'x'. The tangent of the angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is 'x' (the width of the river) and the adjacent side is 100 m (the distance along the river).

So, the equation becomes:

tan(35°) = x/100

To find the value of 'x', we can multiply both sides by 100:

x = 100 * tan(35°)

Using a calculator, we can find that x is approximately 67.1 m. Therefore, the width of the river is approximately 67.1 meters.

Answer 2

Part A:

To find angle D, we make use of the rule of Sines because we know the measure of the side opposite angle D and we also know the meansure another angle of the triangle with the measure of the side opposite the angle.

Thus, we find angle D as follows:


`(62)/(sin(D)) = (51)/(sin(53^o)) \\ \\ \Rightarrow sin(D)= (62sin(53^o))/(51)} \\ \\ = (49.5154)/(51) =0.9709 \\ \\ \Rightarrow D=\sin^(-1){0.9709} \\ \\ =76.14^o=76^o8'`

Therefore, angle D is 76°8'



Part B:

To find angle E, recall that the sum of the angles in a triangle is 180°.

Thus, 53° + 76°8' + E = 180°

E = 180° - 53° - 76°8' = 50°52'

Therefore, angle E is 50°52'



Part C:

To find the measure of side e, we apply the cosine rule as follows:


`e^2=51^2+62^2-2(51)(62)cos(50^o52') \\ \\ =2,601+3,844-2,098.85=3,991.25 \\ \\ \Rightarrow e=√(3,991.25)=63.18`

Therefore, the measure of side e is 63.2



Part D

To find angle G, we make use of the rule of Sines because we know the measure of the side opposite angle G and we also know the measure another angle of the triangle with the measure of the side opposite the angle.

Thus, we find angle G as follows:


`(80)/(sin(G)) = (62)/(sin(49^o)) \\ \\ \Rightarrow sin(G)= (80sin(49^o))/(62) \\ \\ = (60.3768)/(62) =0.9738 \\ \\ \Rightarrow G =\sin^(-1){0.9738}=76.86^o=76^o52'`

Therefore, angle G is 76°52'



Part E:

To find angle H, recall that the sum of the angles in a triangle is 180°.

Thus, 49° + 76°52' + H = 180°

H = 180° - 49° - 76°52' = 54°14'

Therefore, angle H is 54°14'



Part F:

To find the measure of side a, we apply the cosine rule as follows:


`a^2=80^2+62^2-2(80)(62)cos(54^o14') \\ \\ =6,400+3,844-5,798.10=4,445.90 \\ \\ \Rightarrow a=√(4,445.90)=66.68`

Therefore, the measure of side a is 66.7



Part G:

To find angle A, we make use of the rule of Sines because we know the measure of the side opposite angle A and we also know the measure another angle of the triangle with the measure of the side opposite the angle.

Thus, we find angle A as follows:


`(66.7)/(sin(A)) = (30)/(sin(25^o54')) \\ \\ \Rightarrow sin(A)= (66.7sin(25^o54'))/(30) \\ \\ = (29.1347)/(30) =0.9712 \\ \\ \Rightarrow A =\sin^(-1){0.9712}=76.21^o=76^o12'`

Therefore, angle A is 76°12'



Part H:

To find angle B, recall that the sum of the angles in a triangle is 180°.

Thus, 25°54' + 76°12' + B = 180°

B = 180° - 25°54' - 76°12' = 77°53'

Therefore, angle B is 77°53'



Part I:

To find the measure of side b, we make use of the rule of sines

Thus, we find side b as follows:


`(b)/(sin(77^o53')) = (30)/(sin(25^o54')) \\ \\ \Rightarrow b= (30sin(77^o53'))/(sin(25^o54')) \\ \\ = (29.3317)/(0.4368) =67.15`

Therefore, the measure of side b is 67.2