Question

The base of a triangle exceeds the height by 7 feet. if the area is 114 square feet, find the length of the base and the height of the triangle.

Answer 1

Let h represent the height.  Then the base is h+6 (all measurements are in feet).

The formula for the area of a triangle is A = (base)(height)/2.

Here A = 114 ft^2 = (base)(height)/2.  Substituting the (h+6) and h,

A = [114 ft^2] = [(h+6)(h)]/2 , or  [h^2 + 6h]/2, or 2A = 228 = (h+6)h.  Multiply out the right side

Solve 228 = h^2 + 6h for h:

h^2 + 6h - 228 = 0

Applying the Quadratic Formula,
        -6 plus or minus sqrt(36-4(1)(-228))
h = -------------------------------------------------
                              2
               -6 plus or minus sqrt(948)         -6 plus or minus 30.79
or    h = ------------------------------------- = ------------------------------------                                                   2                                                 2
                -6 plus or minus 30.79
        h = ----------------------------------
                               2

We want only the positive result.  That comes to h = 30.79-6
                                                                                    ----------
                                                                                          2

or 24.79/2, or   h = 12. 40

h represents the altitude of the triangle, so h+7 represents the length of the base.  These values are 12.40 feet and 19.40 feet respectively.

Substitute b=19.40 feet and h=12.40 feet into the formula A = bh/2.
Is the result 114 sq. ft. ?