A triangle with area 45 square inches has a height that is three less than three times the base. Find the base and the height of the triangle.

Answer 1

We need to know that formula for area of triangle is A = ¹/₂ hb which means that we have the first of two simultaneous equations in 2 unknowns (h and b)...

(1) 45 = ¹/₂ hb

the second expresses that height is three less than three times the base

(2) h = 3b - 3

We can now substitute h from (2) in (1) to get

45 = ¹/₂ (3b - 3)b

=> 90 = (3b - 3)b = 3b² - 3b

=> 3b² - 3b - 90 = 0

=> (3b + 15)(b - 6) = 0

=> 3b + 15 = 0 or b - 6 = 0

=> b = -5 or b = 6

can't have a negative base, so b must be 6

and if b = 6 then h = 3 x 6 - 3 = 15

Alternatively, you can avoid the quadratic by trying a bit of guesswork:

if ¹/₂ hb = 45 then hb = 90

90 x 1

45 x 2

30 x 3

14 x 5

15 x 6

10 x 9

and we can quickly see that 15 x 6 is the answer as 3 x 6 - 3 = 15

Hence, base = 6 inches and height = 15 inches