The base of a triangle is 3cm grater than the height. The area is 14cm^2. Find the height and length of the base. Please show how you dervied at your answer.

h = x-cm

b = (x + 3) cm
A = 14 cm^2

A = h * b / 2,
x(x + 3) / 2 = 14cm^2
Multiply both sides by 2:,
x(x + 3) = 28,
x^2 + 3x = 28,
x^2 + 3x - 28 = 0,
Factor the trinomial:
(x - 4) (x + 7) = 0,
x + 7 = 0, x = - 7;
x - 4 = 0 , x = 4 cm = h,
x + 3 = 4 + 3 = 7 cm = b

6

Let's call the height of the triangle "h" and the length of the base "b".

From the problem, we know that the base is 3 cm greater than the height, which means:

b = h + 3

We are also given the area of the triangle, which is equal to 14 cm^2. The formula for the area of a triangle is:

Area = (base * height) / 2

Substituting the values given, we have:

14 = (b * h) / 2

Since we know that b = h + 3, we can substitute this into the equation:

14 = ((h + 3) * h) / 2

Multiplying both sides of the equation by 2 to eliminate the fraction, we have:

28 = (h + 3) * h

Expanding the equation, we get:

28 = h^2 + 3h

Rearranging the equation to set it equal to zero, we have:

h^2 + 3h - 28 = 0

To find the values of h that satisfy this equation, we can use the quadratic formula:

h = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 3, and c = -28. Substituting these values into the quadratic formula, we get:

h = (-3 ± √(3^2 - 4(1)(-28))) / (2(1))
h = (-3 ± √(9 + 112)) / 2
h = (-3 ± √121) / 2

Taking the square root of 121, we have:

h = (-3 ± 11) / 2

This gives us two possible values for h:

h = (-3 + 11) / 2 = 8 / 2 = 4

h = (-3 - 11) / 2 = -14 / 2 = -7

Since the height of the triangle cannot be negative, we discard the negative value.

Therefore, the height of the triangle is h = 4 cm.

To find the length of the base, we can substitute the value of h into the equation b = h + 3:

b = 4 + 3
b = 7

Therefore, the length of the base is b = 7 cm.

To find the height and length of the base of the triangle, we can use the formula for the area of a triangle:

Area = 1/2 * base * height

Given that the area is 14cm^2, we can substitute this into the formula:

14 = 1/2 * base * height

Now, let's use the given information that the base is 3cm greater than the height:

base = height + 3

We can substitute this into the formula:

14 = 1/2 * (height + 3) * height

Now, let's simplify the equation:

14 = 1/2 * (height^2 + 3height)

Multiply both sides of the equation by 2 to get rid of the fraction:

28 = height^2 + 3height

Rearrange the equation to make it a quadratic equation by bringing everything to one side:

height^2 + 3height - 28 = 0

Now, we can factorize the quadratic equation:

(height + 7)(height - 4) = 0

Setting each factor equal to zero gives us two possible solutions:

height + 7 = 0 --> height = -7 (discard this solution since height cannot be negative)

height - 4 = 0 --> height = 4

So, the height of the triangle is 4cm.

Now, we can substitute this value back into the equation to find the length of the base:

base = height + 3 = 4 + 3 = 7

Therefore, the height of the triangle is 4cm and the length of the base is 7cm.