Question

In ΔABC, ∠C measures 46° and the values of a and c are 10 and 9, respectively. Find the remaining measurements of the triangle, and round your answers to the nearest tenth. ∠A = 82.2°, ∠B = 62.8°, b = 17.1 ∠A = 53.1°, ∠B = 80.9°, b = 12.4 ∠A = 53.1°, ∠B = 80.9°, b = 17.1 ∠A = 82.2°, ∠B = 62.8°, b = 12.4

Answer 1

Answer:

$\angle{A}=53.1^{\circ}$

$\angle{A}=80.9^{\circ}$

$b=12.4$

Step-by-step explanation:

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We have been given that in ΔABC, ∠C measures 46° and the values of a and c are 10 and 9, respectively.

First of all, we will find measure of angle A using Law Of Sines:

$\frac{\text{sin(A)}}{a}=\frac{\text{sin(B)}}{b}=\frac{\text{sin(C)}}{c}$, where, A, B and C are angles corresponding to sides a, b and c respectively.

$\frac{\text{sin(A)}}{10}=\frac{\text{sin(46)}}{9}$

$\frac{\text{sin(A)}}{10}=\frac{0.719339800339}{9}$

$\frac{\text{sin(A)}}{10}=0.0799266444821111$

$\frac{\text{sin(A)}}{10}*10=0.0799266444821111*10$

$\text{sin(A)}=0.799266444821111$

Upon taking inverse sine:

$A=\text{sin}^{-1}(0.799266444821111)$

$A=53.060109978759^{\circ}$

$A\approx 53.1^{\circ}$

Therefore, the measure of angle A is 53.1 degrees.

Now, we will use angle sum property to find measure of angle B as:

$m\angle{A}+m\angle{B}+m\angle{C}=180^{\circ}$

$53.1^{\circ}+m\angle{B}+46^{\circ}=180^{\circ}$

$m\angle{B}+99.1^{\circ}=180^{\circ}$

$m\angle{B}+99.1^{\circ}-99.1^{\circ}=180^{\circ}-99.1^{\circ}$

$m\angle{B}=80.9^{\circ}$

Therefore, the measure of angle B is 80.9 degrees.

Now, we will use Law Of Cosines to find the length of side b.

$b^2=a^2+c^2-2ac\cdot\text{cos}(B)$

Upon substituting our given values, we will get:

$b^2=10^2+9^2-2(10)(9)\cdot\text{cos}(80.9^{\circ})$

$b^2=100+81-180\cdot 0.158158067254$

$b^2=181-28.46845210572$

$b^2=152.53154789428$

Upon take square root of both sides, we get:

$b=\sqrt{152.53154789428}$

$b=12.3503663060769173$

$b\approx 12.4$

Therefore, the length of side b is approximately 2.4 units.

Answer 2

Answer:

∠A = 53.1°, ∠B = 80.9°, b = 12.4

Step-by-step explanation:

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