Question

18. Your neighbor's yard is in the shape of a triangle, with dimensions 120 ft, 84 ft,

and 85 ft. Is the yard an acute, obtuse, or a right triangle? Explain.

Answer 1

The yard is obtuse!

`\large\underline{\underline{\maltese{\red{\pmb{\sf{\: Explanation :-}}}}}}`

• Determination of acute angled triangle:-When the square of the longest side is less than the sum of the squares of two shorter sides then the triangle is acute angled.
• Determination of right angled triangle:-When the square of the longest side is equal to the sum of the squares of two shorter sides then the triangle is right angled .
• Determination of obtuse angled triangle:-When the square of the longest side is more than the sum of the squares of two shorter sides then the triangle is obtuse angled.

`\red{ \rule{35pt}{2pt}} \orange{ \rule{35pt}{2pt}} \color{yellow}{ \rule{35pt} {2pt}} \green{ \rule{35pt} {2pt}} \blue{ \rule{35pt} {2pt}} \purple{ \rule{35pt} {2pt}}`

Final answer ⤵️

• Longest side = 120 feet
• Shorter side = 85 feet
• Shortest side = 84 feet

`\sf \longrightarrow \: {120}^(2) > {85}^(2) + {84}^(2)`

`\sf \longrightarrow \:14400> 7225 + 7056`

`\sf \longrightarrow \:14400> 14281`

`\qquad \mathbb{TRUE}`

Thus, The triangle is obtuse angled...~

Answer 2

• a=120
• b=84
• c=85

We need angle

Apply law of cosines

`\\ \rm\Rrightarrow c^2=a^2+b^2-2abcos\gamma`

`\\ \rm\Rrightarrow 2abcos\gamma=a^2+b^2-c^2`

`\\ \rm\Rrightarrow cos\gamma=(a^2+b^2-c^2)/(-2ab)`

`\\ \rm\Rrightarrow cos\gamma=(120^2+84^2-85^2)/(-2(120)(84))`

`\\ \rm\Rrightarrow cos\gamma=(14400+7056-7225)/(-20160)`

`\\ \rm\Rrightarrow cos\gamma=(14231)/(-20160)`

`\\ \rm\Rrightarrow cos\gamma=-0.714`

`\\ \rm\Rrightarrow \gamma=cos^(-1)(-0.714)`

`\\ \rm\Rrightarrow \gamma=135.56°`

The yard is obtuse