Question

The length of the base and the height of a triangle are in the ratio 4: 5. If the area of the triangle is 40sq m. Find the lengths of its base and height. ​

Answer 1

Given :

• The length of the base and the height of a triangle are in the ratio 4: 5.
• Area of Triangle is 40 m² .

To Find :

• Length of base and height .

Solution :

$\longmapsto\tt{Let\:Base\:be=4x}$

$\longmapsto\tt{Let\:Height\:be=5x}$

Using Formula :

$\longmapsto\tt\boxed{Area\:of\:Triangle=\dfrac{1}{2}\times{b}\times{h}}$

Putting Values :

$\longmapsto\tt{40=\dfrac{1}{2}\times{4x}\times{5x}}$

$\longmapsto\tt{40\times{2}=20\:{x}^{2}}$

$\longmapsto\tt{80=20{x}^{2}}$

$\longmapsto\tt{{x}^{2}=\cancel\dfrac{80}{20}}$

$\longmapsto\tt{{x}^{2}=4}$

$\longmapsto\tt\bf{2=x}$

Value of x is 2 .

Therefore :

$\longmapsto\tt{Length\:of\:Base=4(2)}$

$\longmapsto\tt\bf{8\:m}$

$\longmapsto\tt{Length\:of\:Height=5(2)}$

$\longmapsto\tt\bf{10\:m}$

Answer 2

Given :

• Area = 40 cm²
• Ratio of Base and Height = 4:5

To Find :

• Base and Height = ?

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SolutioN :

$\maltese$ Formula Used :

• ${\underline{\boxed{\pmb{\sf{ Area{\small_{(Triangle)}} = \dfrac{1}{2} \times Base \times Height }}}}}$

$\maltese$ Let the Ratios :

• Base = 4y
• Height = 5y

$\maltese$ Deriving the Value of y :

$\begin{gathered} \qquad \; \dashrightarrow \; \; \sf { Area = \dfrac{1}{2} \times Base \times Height } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \dashrightarrow \; \; \sf { 40 = \dfrac{1}{2} \times 4y \times 5y } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \dashrightarrow \; \; \sf { 40 = \dfrac{1}{2} \times {20y}^{2} } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \dashrightarrow \; \; \sf { 40 = \dfrac{1}{\cancel2} \times \cancel{20y}^{2} } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \dashrightarrow \; \; \sf { 40 = 1 \times {10y}^{2} } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \dashrightarrow \; \; \sf { \dfrac{40}{10} = {y}^{2} } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \dashrightarrow \; \; \sf { \cancel\dfrac{40}{10} = {y}^{2} } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \dashrightarrow \; \; \sf { 4 = {y}^{2} } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \dashrightarrow \; \; \sf { \sqrt{4} = y } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \dashrightarrow \; \; {\underline{\boxed{\pmb{\frak{ y = 2 }}}}} \; {\purple{\pmb{\bigstar}}} \\ \\ \\ \end{gathered}$

$\maltese$ Calculating the Dimensions :

• Base = 4y = 4(2) = 8 cm
• Height = 5y = 5(2) = 10 cm

$\therefore \;$ Base of the Triangle is 8 cm and its Height is 10 cm .

$\\ \qquad{\rule{200pt}{2pt}}$