Question

the height of a triangle is 2 cm more than the base. if the height is increased by 1 cm while the base remains the same, the new area becomes 14 cm²
(a) find the base and height of the original triangle.
(b) find the area of the original triangle.
part 0/3
part 1 of 3
the original triangle had a base of □ cm.

Explanation:

Step1: Set up variables

Let the base of the original triangle be \(x\) cm. Then the height of the original triangle is \((x + 2)\) cm.

Step2: Write the area formula for the new triangle

The area of a triangle is \(A=\frac{1}{2}\times base\times height\). The base of the new triangle is still \(x\) cm and the height of the new triangle is \((x + 2+1)=(x + 3)\) cm. We know that the area of the new triangle \(A = 14\) \(cm^{2}\). So, \(\frac{1}{2}\times x\times(x + 3)=14\).

Step3: Solve the quadratic equation

Multiply both sides of the equation \(\frac{1}{2}x(x + 3)=14\) by \(2\) to get \(x(x + 3)=28\). Expand it: \(x^{2}+3x-28 = 0\). Factor the quadratic equation: \(x^{2}+3x-28=(x + 7)(x - 4)=0\).

Step4: Find the value of \(x\)

Set each factor equal to zero: \(x+7 = 0\) gives \(x=-7\) (rejected since length cannot be negative), \(x - 4=0\) gives \(x = 4\).

Answer:

The original triangle had a base of \(4\) cm.