Respuesta :
the length of the shorter leg = 10m
Explanation:
longer leg = 24m
shorter leg = ?
hypotenuse = 6 more than twice the shorter leg
[tex]\text{hypotenuse = 6 + 2(shorter leg)}[/tex]Since the triangle is a right angled triangle, to get the shorter leg we will apply pythagoras theorem:
Hypotenuse² = opposite² + adjacent²
Hypotenuse² = (shorter leg)² + (longer leg)²
substitute the values in the formula:
[tex]\begin{gathered} \lbrack6+2(shorter\text{ }leg)\rbrack^2\text{ = (shorter leg)}^2+(\text{24})^2 \\ \text{let x represent shorter leg} \\ \lbrack6+2(x)\rbrack^2\text{ = (x)}^2+(\text{24})^2 \\ \lbrack6+2x\rbrack^2\text{ = x}^2+24^2 \end{gathered}[/tex][tex]\begin{gathered} (6+2x\rbrack)(6+2x)\text{ = x}^2+576 \\ 6(6\text{ + 2x) + 2x(6 + 2x) = x}^2+576 \\ 36+12x+12x+4x^2\text{ = x}^2+576 \\ \text{collect like terms:} \\ 4x^2\text{- x}^2+36+12x+12x=576 \end{gathered}[/tex][tex]\begin{gathered} 3x^2+36+24x=576 \\ 3x^2+24x+36-576\text{ = 0} \\ 3x^2+24x-540\text{ = 0} \\ \text{divide through by 3:} \\ x^2+8x-180\text{ = 0} \end{gathered}[/tex]To get x, we will apply factorisation method:
[tex]\begin{gathered} x^2\text{ + 18x - 10x - 180 = 0} \\ x(x\text{ + 18) -10(x + 18) = 0} \\ (x\text{ - 10)(x + 18) = 0} \\ x\text{ - 10 = 0 or x + 18 = 0} \\ x\text{ = 10 or x = -18} \end{gathered}[/tex]Since we can't have a negative length, the value of x will be 10
As a result, the length of the shorter leg = 10m
