Question

the length of the sides forming a right angle of a right-angled triangle is 5 x cm and 3 X - 1 cm. area of the triangle is 60 CM square. find the hypotenuse

Answer 1

hypotenuse (h) = 17 cm

using area of a triangle formula to solve for x

A =
`(1)/(2)` bh ( b is the base and h the height )

`(1)/(2)` × 5x(3x - 1 ) =60 ( multiply through by 2 )

5x(3x - 1)=120

15x² - 5x - 120 = 0 ← in standard form ( divide all terms by 5 )

3x² - x - 24 = 0

consider the factors of the product 3 × - 24 = - 72 which sum to - 1

The factors are - 9 and + 8 ( split the middle term using these factors )

3x² - 9x + 8x - 24 = 0 ( factor by grouping )

3x(x - 3) + 8(x - 3) = 0 ( take out common factor (x - 3) )

(x - 3 )(3x + 8) = 0 ( equate each factor to zero and solve for x )

x - 3 = 0 ⇒ x = 3

3x + 8 = 0 ⇒ x = -
`(8)/(3)`

x > 0 ⇒ x = 3

the sides are 5x = 15 and 3x - 1 = 8

h = √(15² + 8² ) = √(225 + 64 ) = 17 ← ( hypotenuse )