Law of Sines Ambiguous Case: How Many Triangles?

1) In △ABC, side a = 8, side b = 12, and angle A = 40°. How many triangles can be formed?

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Infinitely many

h=b·sinA≈7.71; since h

Explanation

h=b·sinA≈7.71; since h

2) In △ABC, side a = 5, side b = 10, and angle A = 40°. How many triangles can be formed?

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Infinitely many

h≈6.43; a

Explanation

h≈6.43; a

3) In △ABC, side a = 12, side b = 7, and angle A = 50°. How many triangles can be formed?

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Infinitely many

h≈5.36; a≥b so exactly one triangle.

Explanation

h≈5.36; a≥b so exactly one triangle.

4) In △ABC, side a = 10, side b = 20, and angle A = 30°. How many triangles can be formed?

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Infinitely many

h=b·sin30°=10; a=h ⇒ one (right) triangle.

Explanation

h=b·sin30°=10; a=h ⇒ one (right) triangle.

5) In △ABC, side a = 9, side b = 4, and angle A = 40°. How many triangles can be formed?

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Infinitely many

h≈2.57; a≥b so exactly one triangle.

Explanation

h≈2.57; a≥b so exactly one triangle.

6) In △ABC, side a = 15, side b = 10, and angle A = 50°. How many triangles can be formed?

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Infinitely many

h≈7.66; a≥b so exactly one triangle.

Explanation

h≈7.66; a≥b so exactly one triangle.

7) In △ABC, side a = 7, side b = 7, and angle A = 60°. How many triangles can be formed?

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Infinitely many

h≈6.06; a≥b so exactly one triangle.

Explanation

h≈6.06; a≥b so exactly one triangle.

8) In △ABC, side a = 6, side b = 12, and angle A = 40°. How many triangles can be formed?

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Infinitely many

h≈7.71; a

Explanation

h≈7.71; a

9) In △ABC, side a = 4, side b = 9, and angle A = 30°. How many triangles can be formed?

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Infinitely many

h=9·sin30°=4.5; a

Explanation

h=9·sin30°=4.5; a

10) In △ABC, side a = 10, side b = 10, and angle A = 45°. How many triangles can be formed?

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Infinitely many

h≈7.07; a≥b so exactly one triangle.

Explanation

h≈7.07; a≥b so exactly one triangle.

11) In △ABC, side a = 3, side b = 8, and angle A = 30°. How many triangles can be formed?

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Infinitely many

h=8·sin30°=4; a

Explanation

h=8·sin30°=4; a

12) In △ABC, side a = 20, side b = 15, and angle A = 40°. How many triangles can be formed?

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Infinitely many

h≈9.64; a≥b so exactly one triangle.

Explanation

h≈9.64; a≥b so exactly one triangle.

13) In △ABC, side a = 12, side b = 12, and angle A = 80°. How many triangles can be formed?

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Infinitely many

h≈11.82; a≥b so exactly one triangle.

Explanation

h≈11.82; a≥b so exactly one triangle.

14) In △ABC, side a = 8, side b = 16, and angle A = 50°. How many triangles can be formed?

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Infinitely many

h≈12.26; a

Explanation

h≈12.26; a

15) In △ABC, side a = 9, side b = 9, and angle A = 90°. How many triangles can be formed?

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Infinitely many

A=90° and a=b ⇒ B=90° (degenerate). Only possible if a>b; here none.

Explanation

A=90° and a=b ⇒ B=90° (degenerate). Only possible if a>b; here none.

16) In △ABC, side a = 10, side b = 25, and angle A = 20°. How many triangles can be formed?

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Infinitely many

h≈8.55; h

Explanation

h≈8.55; h

17) In △ABC, side a = 14, side b = 10, and angle A = 40°. How many triangles can be formed?

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Infinitely many

h≈6.43; a≥b so exactly one triangle.

Explanation

h≈6.43; a≥b so exactly one triangle.

18) In △ABC, side a = 5, side b = 9, and angle A = 80°. How many triangles can be formed?

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Infinitely many

h≈8.86; a

Explanation

h≈8.86; a

19) In △ABC, side a = 15, side b = 30, and angle A = 60°. How many triangles can be formed?

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Infinitely many

h≈25.98; a

Explanation

h≈25.98; a

20) In △ABC, side a = 7, side b = 3, and angle A = 40°. How many triangles can be formed?

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Infinitely many

h≈1.93; a≥b so exactly one triangle.

Explanation

h≈1.93; a≥b so exactly one triangle.

Determining the Number of Possible Triangles (Ambiguous Case)