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.