The base of a triangular garden is 5 yards longer than the​ height, and the area of the garden is 168 square yards. Find the dimensions of the triangle?

Let the height of the triangular shaped garden be h yards. Its base would then be 5+h yards.

Formula for the area of any traingle is #1/2# base X height.

By this formula, #1/2# (5+h) Xh= 168

On simplifying this. #h^2 +5h =336#

Or #h^2 +5h-336=0#. Now factorise this quadratic expression by splitting the middle term, keeping in view the fact that 21X16=336 and difference 21-16 =5. This would be as follows:

#h^2 +21h -16h -336=0#. Now pair these terms,

#(h^2 +21h) -(16h +336)=0#
h(h+21) -16(h+21)=0

(h+21)(h-16)=0. This means either h-16=0 or h+21=0.

Since h+21=0 would mean h= -21, this cannot be the required solution, because height of the triangle can never be negative. Hence, h-16=0 would give the height of the triangle as h= 16 yards. The dimensions of the triangle would be height = 16 yards and length of the base = 21 yards
*****

#"let the height "=h#

#"then base "=h+5larrcolor(blue)"5 yards longer"#

#• " area of triangle "=1/2" base "xx"height"#

#rArr1/2h(h+5)=168#

#"multiply both sides by 2"#

#rArrh(h+5)=336#

#"express in standard form and factorise"#

#rArrh^2+5h-336=0larrcolor(blue)"in standard form"#

#"the factors of - 336 which sum to + 5 are + 21 and - 16"#

#rArr(h+21)(h-16)=0#

#"equate each factor to zero and solve for h"#

#h+21=0rArrh=-21larrcolor(red)"not valid"#

#h-16=0rArrh=16#

#"since "h>0" then h = 16"#

#"base of triangle "=h+5=16+5=21" yards"#

#"and height "=h=16" yards"#

#color(blue)"As a check"#

#"area "=1/2bh=1/2xx21xx16=168" square yards"#