Question

PLEASE HELP DUE SOON! PLEASE SHOW WORK SO I KNOW HOW TO IT FOR NEXT TIME! IM STUCK ON THIS AND ITS DUE TODAY!!!!! YOU WILL GETT 100 POINTS REAL ANSWERS ONLY! MANY THANKS! QUESTIONS DOWN BELOW!!!!!

Answer 1

a) zero triangles.

b) one triangle.

Explanation:

In triangle ABC:

• A, B and C are the interior angles.
• a, b and c are the sides opposite the corresponding interior angles.

Part (a)

Given:

• ∠A = 51°
• a = 10 cm
• b = 28 cm

Law of Sines

`\sf (\sin A)/(a)=(\sin B)/(b)=(\sin C)/(c)`

(where A, B and C are the angles and a, b and c are the sides opposite the angles)

To determine if any triangles are possible, substitute the given values into the Law of Sines to find angle B:

`\implies \sf (\sin 51^(\circ))/(10)=(\sin B)/(28)`

`\implies \sf \sin B=(28\sin 51^(\circ))/(10)`

`\implies \sf \sin B=2.176008...`

As -1 ≤ sin B ≤ 1, there is no solution for angle B.

Therefore, zero triangles are possible.

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Part (b)

Given:

• ∠C = 30°
• a = 24 cm
• c = 12 cm

Law of Sines

`\sf (\sin A)/(a)=(\sin B)/(b)=(\sin C)/(c)`

(where A, B and C are the angles and a, b and c are the sides opposite the angles)

To determine if any triangles are possible, substitute the given values into the Law of Sines to find angle A:

`\implies \sf (\sin A)/(24)=(\sin 30^(\circ))/(12)`

`\implies \sf \sin A=(24\sin 30^(\circ))/(12)`

`\implies \sf \sin A=1`

`\implies \sf A=\sin^(-1)(1)`

`\implies \sf A=90^(\circ)`

Therefore, one triangle is possible (see attachment).