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,

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)} $$