Question

A triangle with a base of 8 meters has an area of 60 square meters.The longest side of the triangle is 17 meters.Is the triangle a right triangle?

Answer 1

Yes, the triangle is a right triangle.

Explanation:

A triangle with a base of 8 meters and longest side is 17 meters.

Area of triangle is
`60\text{ m}^2.`

Let third side be x m

Semi-perimeter of triangle,
`s=(8+17+x)/(2)=(25+x)/(2)`

Using Heron's formula to find area of triangle.

`\text{Area of triangle }=√(s(s-a)(s-b)(s-c))`

`60=\sqrt{(25+x)/(2)\left((25+x)/(2)-8\right)\left((25+x)/(2)-17\right)\left((25+x)/(2)-x\right)}`

squaring both sides

`3600=(25+x)/(2)* (x+9)/(2)* (x-9)/(2)* (25-x)/(2)`

`3600=(-50625+706x^(2)-x^(4))/(16)`

`x^4-706x^2+108225=0`

`(x^2-225)(x^2-481)=0`

`x^2=225\text{ or }x^2=481`

`x=\pm 15,\pm√(481)`

x should be positive integer.

So,
`x=15\text{ m}`

Using pthagoreous theorem:

`17^2=15^2+8^2`

`289=225+64`

`289=289`

Therefore, the given triangle is a right triangle.