Right triangle has a hypotenuse with length 41 cm. The area of the triangle is 180 cm^2. What are the triangle leg lengths?

1 Answer
Feb 21, 2018

#40" cm and "9" cm"#

Explanation:

#"area of triangle "=1/2bh#

#"where b is the base and h the height"#

#"expressing b in terms of h"#

#"using "color(blue)"Pythagoras' theorem"#

#b=sqrt(41^2-h^2)#

#"now area "=180#

#rArr1/2bh=180rArrbh=360#

#rArrhsqrt(41^2-h^2)=360#

#color(blue)"square both sides"#

#rArrh^2(41^2-h^2)=360^2#

#rArr1681h^2-h^4=129600#

#"multiply through by "-1" and equate to zero"#

#rArrh^4-1681h^2+129600=0#

#"use the substitution "u=h^2#

#rArru^2-1681u+129600=0#

#"solve using the "color(blue)"quadratic formula"#

#u=(1681+-sqrt2307361)/2=(1681+-1519)/2#

#u=(1681+1519)/2=1600" or "u=(1681-1519)/2=81#

#u=h^2rArrh^2=1600" or "h^2=81rArrh=40" or "h=9#

#h=9rArrb=360/9=40#

#"h=40rArrb=360/40=9#

#color(blue)"As a check"#

#h=40,b=9rArrsqrt(40^2+9^2)=41#

#"and area "=1/2xx40xx9=180" cm"^2#

#rArr"lengths of legs "=40" and "9 "cm"#