Question

How do you find the height of a scalene triangle when given 2 angles and 1 side: side c = 1200, Angle A = 72, Angle B = 77?

Answer 1

Hint: Here in this question, we have to find the height of the scalene triangle. Firstly, find the third angle c using the Interior Angles of a Triangle Rule, using the 3 angle we can easily find the all sides of triangle Using the sine and cosine rules i.e., \[\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}\] later by using the Heron’s formula \[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \] on substituting this in the formula of area of triangle \[A = \dfrac{1}{2} \times base \times height\], on simplification we can get the required height of triangle.

Complete step by step solution:
Scalene Triangle is a triangle that has all its sides of different lengths. It means all the sides of a scalene triangle are unequal and all the three angles are also of different measures.
Given the angle \[\left| \!{\underline {\,
  A \,}} \right. = {72^0}\], \[\left| \!{\underline {\,
  B \,}} \right. = {77^0}\] and the length of c is 1200
Now find the angle \[\left| \!{\underline {\,
  C \,}} \right. \]
By interior Angles of a Triangle, the sum of all 3 interior angles in a triangle is \[{180^0}\] i.e,
\[ \Rightarrow \,\,\,\left| \!{\underline {\,
  A \,}} \right. + \left| \!{\underline {\,
  B \,}} \right. + \left| \!{\underline {\,
  C \,}} \right. = {180^0}\]
\[ \Rightarrow \,\,\,{72^0} + {77^0} + \left| \!{\underline {\,
  C \,}} \right. = {180^0}\]
\[ \Rightarrow \,\,\,{149^0} + \left| \!{\underline {\,
  C \,}} \right. = {180^0}\]
\[ \Rightarrow \,\,\,\left| \!{\underline {\,
  C \,}} \right. = {180^0} - {149^0}\]
\[ \Rightarrow \,\,\,\left| \!{\underline {\,
  C \,}} \right. = {31^0}\]
Now, using the angles \[\left| \!{\underline {\,
  A \,}} \right. \], \[\left| \!{\underline {\,
  B \,}} \right. \] , \[\left| \!{\underline {\,
  C \,}} \right. \] and the length of c, we can find the length of another sides by using the sine and cosine rule i.e., \[\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}\]
\[ \Rightarrow \,\,\,\dfrac{a}{{\sin {{72}^0}}} = \dfrac{b}{{\sin {{77}^0}}} = \dfrac{{1200}}{{\sin {{31}^0}}}\]
On rearranging we can written a and b as
\[ \Rightarrow \,\,\,a = \dfrac{{1200 \times \sin {{72}^0}}}{{\sin {{31}^0}}}\] and \[b = \dfrac{{1200 \times \sin {{77}^0}}}{{\sin {{31}^0}}}\]
\[ \Rightarrow \,\,\,a = \dfrac{{1200 \times 0.951}}{{0.515}}\] and \[b = \dfrac{{1200 \times 0.974}}{{0.515}}\]
On simplification we get
\[ \Rightarrow \,\,\,a = 2,215.92\] and \[b = 2,269.51\].
Now find the area of triangle using Heron’s formula i.e.,
\[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \]
Where ‘s’ is the semi-perimeter which can be find by the formula i.e., \[s = \dfrac{{a + b + c}}{2}\]
\[ \Rightarrow \,s = \dfrac{{2,215.92 + 2,269.51 + 1200}}{2}\]
\[ \Rightarrow \,s = \dfrac{{5,685.43}}{2}\]
\[ \Rightarrow \,s = 2,842.715\]
Substitute the value s, a, b and c in Heron’s formula, we get
 \[ \Rightarrow \,\,A = \sqrt {2,842.715\left( {2,842.715 - 2,215.92} \right)\left( {2,842.715 - 2,269.51} \right)\left( {2,842.715 - 1200} \right)} \]
\[ \Rightarrow \,\,A = \sqrt {2,842.715\left( {626.795} \right)\left( {573.205} \right)\left( {1,642.715} \right)} \]
\[ \Rightarrow \,\,A = \sqrt {1,677,764,641,007.7} \]
\[ \Rightarrow \,\,A = 12,95,285.544\]
Now, to find the height of triangle by using the formula of area of triangle i.e., \[A = \dfrac{1}{2} \times base \times height\]
On rearranging this
\[ \Rightarrow \,\,\,height = \dfrac{{2 \times A}}{{base}}\]
\[ \Rightarrow \,\,\,height = \dfrac{{2 \times 12,95,285.544}}{{1200}}\]
\[ \Rightarrow \,\,\,height = 2158.81\]
Hence, the height of the given scalene triangle is \[2158.81\].
So, the correct answer is “ \[2158.81\] Units”.

Note: In triangle we have 3 different kinds. This is classified based on the lengths of triangles. We know by interior Angles of a Triangle, the sum of all 3 interior angles in a triangle is \[{180^0}\] . The area of triangle using Heron’s formula i.e.,\[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \]