Math  /  Geometry

QuestionGiven the measurements of a triangle: mB=110 m \angle B = 110^\circ , b=11m b = 11 \, \text{m} , and a=4m a = 4 \, \text{m} , determine the number of possible triangles that can be formed with these measurements.

Studdy Solution

STEP 1

1. We are given a triangle with angle B=110 B = 110^\circ .
2. The side opposite angle B B is b=11m b = 11 \, \text{m} .
3. The side a=4m a = 4 \, \text{m} is opposite angle A A .
4. We need to determine the number of possible triangles that can be formed with these measurements.

STEP 2

1. Use the Law of Sines to determine if a triangle can be formed.
2. Analyze the possible cases for the triangle formation.
3. Determine the number of possible triangles.

STEP 3

Use the Law of Sines, which states:
asinA=bsinB \frac{a}{\sin A} = \frac{b}{\sin B}
Substitute the given values:
4sinA=11sin110 \frac{4}{\sin A} = \frac{11}{\sin 110^\circ}

STEP 4

Calculate sin110 \sin 110^\circ :
sin110=sin(18070)=sin70 \sin 110^\circ = \sin (180^\circ - 70^\circ) = \sin 70^\circ
Use a calculator to find sin700.9397 \sin 70^\circ \approx 0.9397 .

STEP 5

Substitute sin1100.9397 \sin 110^\circ \approx 0.9397 into the equation:
4sinA=110.9397 \frac{4}{\sin A} = \frac{11}{0.9397}
sinA=4×0.939711 \sin A = \frac{4 \times 0.9397}{11}
sinA3.758811 \sin A \approx \frac{3.7588}{11}
sinA0.3417 \sin A \approx 0.3417

STEP 6

Determine if sinA0.3417 \sin A \approx 0.3417 is valid:
Since sinA \sin A is a valid sine value (between -1 and 1), angle A A can exist. Use the inverse sine function to find angle A A :
A=sin1(0.3417) A = \sin^{-1}(0.3417)
Calculate A20 A \approx 20^\circ .

STEP 7

Check the sum of angles in the triangle:
Since A20 A \approx 20^\circ and B=110 B = 110^\circ , find angle C C :
C=18011020 C = 180^\circ - 110^\circ - 20^\circ
C=50 C = 50^\circ

STEP 8

Since all angles are positive and their sum is 180 180^\circ , a triangle can be formed. Check for any other possible triangles:
In this case, since angle B B is obtuse (110 110^\circ ), there can only be one triangle formed. An obtuse angle in a triangle ensures that no other configuration can satisfy the triangle inequality.
The number of possible triangles that can be formed with these measurements is:
1 \boxed{1}

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