7. The sides of a right triangular park are in the ratio of 3 : 4, while its area is 486 sq m. Find the length of its hypotenuse.
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7. The sides of a right triangular park are in the ratio of 3 : 4, while its area is 486 sq m. Find the length of its hypotenuse.
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Given :
To Find :
Solution :
According to the question :
Let, the sides of the park be 3x and 4x
Using Formula : Area of triangle = base × height
[tex]\longrightarrow [/tex] 486 = 1/2 × 3x × 4x
[tex]\longrightarrow [/tex] 486 = 1/2 × 12x²
[tex]\longrightarrow [/tex] 486 = 6x²
[tex]\longrightarrow [/tex] x² = 486/6
[tex]\longrightarrow [/tex] x² = 81
[tex]\longrightarrow [/tex] x = √81
[tex]\longrightarrow [/tex] x = 9
Reference of Figure :
[tex]\setlength{\unitlength}{0.99cm}\begin{picture}(6, 4)\linethickness{0.26mm}\put(1,2){\line(1,0){2.8}}\put(3.9,4){\sf{36m}}\put(3.8,2){\line(0,2){4.5}}\put(2,1.7){\sf{27m}}\qbezier(1,2.05)(1.4,3)(3.8,6.5)\put(0.7,1.7){\bf{A}}\put(3.9,1.7){\bf{B}}\put(3.7,6.6){\bf{C}}\end{picture}[/tex]
[tex]\longrightarrow [/tex] AB = 3x = 3(9) = 27
[tex]\longrightarrow [/tex] BC = 4x = 4(9) = 36
Using Pythagoras Theorem : AC² = AB² + BC²
[tex]\longrightarrow [/tex] AC² = (27)² + (36)²
[tex]\longrightarrow [/tex] AC² = 729 + 1296
[tex]\longrightarrow [/tex] AC² = 2025
[tex]\longrightarrow [/tex] AC = √2025
[tex]\longrightarrow [/tex] AC = 45m
Hence, the hypotenuse of the park is 45m.