Question #142869
Solve the following triangles. Identify the case # in each given
triangle.

6.) Given parts:

a= 173 b= 267 c=412
1
Expert's answer
2020-11-10T18:55:41-0500

Here is the sketch of the triangle;


We are given all the three sides. We need to calculate the three angles A, B, and C.

Here we apply cosine rule.

a2=b2+c22bccosAa^2 = b^2+c^2 - 2bccosA

And this can be re-written as;

cosA=b2+c2a22bccosA = \frac{b^2 + c^2 - a^2}{2bc}


cosA=2672+412217322×267×412cos A = \frac{267^2+ 412^2 - 173^2}{2 × 267 × 412}


=71289+16974429929220008=\frac{71289+169744-29929}{220008}


=211104220008=\frac{211104}{220008}


=0.95952874441= 0.95952874441


cosA=0.95952874441\therefore cosA =0.95952874441

A=cos1(0.95952874441)A = cos^{-1} (0.95952874441)

=16.36°= 16.36 °


To find angle B, we again apply cosine rule

b2=a2+c22accosBb^2= a^2 +c^2 - 2ac cos B

Which can also be re-written;

cosB=a2+c2b22accos B= \frac{a^2+c^2-b^2}{2ac}


cosB=1732+412226722×173×412cos B= \frac{173^2+412^2-267^2}{2 × 173 × 412}


=29929+16974471289142552=\frac{29929+169744-71289}{142552}


=128384142552=\frac{128384}{142552}


=0.90061170661=0.90061170661


B=cos1(0.90061170661)B = cos^{-1}(0.90061170661)


B=25.76°B= 25.76 °


To find the final angle C, we apply the property: 'The sum of the three interior angles of a triangle add up to 180°'


C=180°(A+B)\therefore C = 180 ° - (A+B)


=180°(16.36°+25.76°)= 180 ° - (16.36 ° + 25.76 °)

=180°42.12°=180 ° - 42.12 °

=137.88°=137.88 °


Therefore, the three angles are;

A=16.36°A= 16.36 °

B=25.76°B = 25.76 °

C=137.88°C=137.88 °


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