The base of a triangle is 3 cm greater than the height. The area is 14cm^2. Find the length and height of the base.

let the height be x cm

then the base is x+3 cm

x(x+3)/2 = 14

solve for x

Let's assume the height of the triangle is h cm. According to the given information, the base of the triangle is 3 cm greater than the height. So, the length of the base is (h + 3) cm.

The formula for the area of a triangle is given by:

Area = (1/2) * base * height

Substituting the given values, we can write the equation:

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

To solve this equation, let's multiply both sides by 2 to get rid of the fraction:

28 = (h + 3) * h

Expanding the right side:

28 = h^2 + 3h

Rearranging the equation in standard quadratic form:

h^2 + 3h - 28 = 0

Now, we can solve this quadratic equation using factoring or the quadratic formula. Factoring the equation, we find:

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

Setting each factor equal to zero, we get two possible solutions:

h + 7 = 0 or h - 4 = 0

If h + 7 = 0, then h = -7. However, since the height cannot be negative in this context, we can ignore this solution.

If h - 4 = 0, then h = 4.

Therefore, the height of the triangle is 4 cm.

Now, we can find the length of the base by substituting the value of h back into our expression for the length of the base:

Length of base = h + 3 = 4 + 3 = 7 cm.

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

To find the length and height of the base, we need to solve the following problem step by step.

Step 1: Understand the problem
We have a triangle with a base that is 3 cm longer than the height. The area of the triangle is given as 14cm^2.

Step 2: Identify the known values
Let's assign variables to the unknown values.
Let h represent the height of the triangle.
Then the base of the triangle is h + 3 cm.

Step 3: Use the area formula for a triangle
The area of a triangle is given by the formula:
Area = (1/2) * base * height

We know that the area is 14 cm^2, so we have:
14 = (1/2) * (h + 3) * h

Step 4: Simplify the equation
Multiply both sides of the equation by 2 to get rid of the fraction:
28 = (h + 3) * h

Step 5: Expand the equation
28 = h^2 + 3h

Step 6: Rearrange the equation
Rearrange the equation in standard quadratic form (ax^2 + bx + c = 0):
h^2 + 3h - 28 = 0

Step 7: Solve the equation
We can solve this quadratic equation by factoring or using the quadratic formula. Let's use factoring:

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

Setting each factor equal to zero, we get:
h + 7 = 0 or h - 4 = 0

Solving for h, we find:
h = -7 or h = 4

Since the height of a triangle cannot be negative, we discard the negative value and conclude that:
h = 4 cm

Step 8: Find the base length
To find the length of the base, we substitute the value of h into the equation:
base = h + 3
base = 4 + 3
base = 7 cm

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

The area of a triangle is

A = base * height /2

If you have not learned to solve quadratic equations before, then think of
2A = base * height
Since A=14, 2A = 28
Look for two integers that multiply together to give 28, but with a difference of 3 between them.

If you have learned quadratic equations, then let
h=height
b=base = h+3

Using the area formula above, you'll get
A=14 = h(h+3)/2
from which you can solve for the height h. The base is simply h+3.