The base of a triangle is 3 cm greater than the height. The area is 27 cm squared. Find the height and base of the triangle.
Let's represent the height of the triangle as 'h' cm.
According to the given information, the base is 3 cm greater than the height. Therefore, the base of the triangle can be represented as 'h + 3' cm.
The area of a triangle can be calculated using the formula:
Area = (1/2) * base * height
Substituting the given values into the formula, we have:
27 = (1/2) * (h + 3) * h
To solve for the height, we need to rearrange the equation and solve for 'h':
27 = (1/2) * (h^2 + 3h)
Multiplying both sides by 2 to eliminate the fraction:
54 = h^2 + 3h
Rearranging the equation and setting it equal to zero:
h^2 + 3h - 54 = 0
We can now solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's solve it by factoring:
(h + 9)(h - 6) = 0
Setting each factor equal to zero:
h + 9 = 0 or h - 6 = 0
Solving for 'h' in each case:
h = -9 or h = 6
Since the height of a triangle cannot be negative, we can discard h = -9.
Therefore, the height of the triangle is 6 cm.
Substituting this value back into one of the earlier equations for the base:
base = height + 3 = 6 + 3 = 9 cm
So, the height of the triangle is 6 cm and the base is 9 cm.
To find the height and base of the triangle, we can use the formula for the area of a triangle:
Area = (base * height) / 2
Given that the area is 27 cm² and the base is 3 cm greater than the height, we can set up the following equation:
27 = (height * (height + 3)) / 2
Let's solve this equation step-by-step to find the values of the height and base.
Step 1: Multiply both sides of the equation by 2 to eliminate the fraction:
27 * 2 = height * (height + 3)
Step 2: Simplify both sides of the equation:
54 = height^2 + 3height
Step 3: Rearrange the equation:
height^2 + 3height - 54 = 0
Step 4: Factorize the quadratic equation:
(height - 6)(height + 9) = 0
Now we have two possible solutions for the height:
1. height - 6 = 0 --> height = 6
2. height + 9 = 0 --> height = -9
Since the height of a triangle cannot be negative, we discard the second solution. Therefore, the height of the triangle is 6 cm.
To find the base, we use the given information that the base is 3 cm greater than the height. So, the base is:
base = height + 3 = 6 + 3 = 9 cm
Therefore, the height of the triangle is 6 cm and the base is 9 cm.
bh/2 = 27
(h+3)h/2 = 27
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