2 centimeters 7

Question

The area of a triangular flag is
56
square centimeters. Its altitude is 2 centimeters longer than twice its base. Find the lengths of the altitude and the base.

Answer 1

 

The area of a triangle is:

 

    Area = (1/2) • Base • Altitude

 

Let

            b = base of triangle

     2b+2 = altitude of triangle ... (2 cm longer than twice the base)

 

So we have

 

     56 = (1/2) • b • (2b+2)

     56 = (2b² + 2b)/2

     56 = 2(b² + b)/2

     56 = b² + b

     0 = b² + b - 56

     b² + b - 56 = 0

     (b - 7)(b + 8) = 0

 

Using zero product property,

 

     b - 7 = 0 ... b =7

     or

     b + 8 = 0 ... b = -8 .... discard this, we can't have a negative side measure

 

So the base is 7

The altitude is 2b + 2 = 2(7) + 2 = 14 + 3 = 16

 

CHECK:

 

  AREA = 1/2 • base • height

   56 = 1/2 • 7 • 16

   56 = 56

 

Answer 2

Jessica,

 

Let x be the length of the base. the length of the altitude is x + 2.

 

The area of a triangle is ½ times the base times the height. If this equals 56 cm², then you can set up an equation

 

½ x(x + 2) = 56.

 

The idea is to solve for x, then find x + 2.

 

Solving for x, you first multiply both sides by 2 (to cancel the fraction), then distribute on the left side:

 

x(x + 2) = 112

 

x² + 2x = 112

 

x² + 2x − 112 = 0.

 

Factoring this requires finding two numbers whose product is −112 and whose sum is 2. This is quote difficult to do, so your next options are to complete the square or use the quadratic formula. I will use the latter.

 

a = 1, b = 2, c = −112  plugged into x = (−b ± √(b² − 4ac))/(2a)  produces

 

x = (−2 ± √(2² + 4·112))/2 = (−2 ± √452)/2 = (−2 ± 2√113)/2 = −1 ± √113

 

As you can see, the equation has two answers, but one of them is negative so it can be safely ignored. (That's the −1 − √113 solution.)

 

Using the positive solution,

 

x = −1 + √113  and x + 2 = 1 + √113.

 

The answers are: the base is −1 + √113 and the altitude has length 1 + √113 . If you want these in decimal form, you may use a calculator to round the answers as needed. Both lengths (of course) are in centimeters.