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"