The area of a triangle is 66 m2, and its base is 1 m more than the height. Find the base and height of the triangle.
(1/2) b h = 66 so b h = 33
b = h + 1
so
(h+1)(h) = 33
h^2 + h - 33 = 0
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then b = h + 1
Let's assume the height of the triangle is 'h' meters. According to the given information, the base of the triangle is '1' meter more than the height. So, the length of the base would be 'h + 1'.
The formula to calculate the area of a triangle is:
Area = (1/2) * base * height
Given that the area of the triangle is 66 m², we can substitute the values into the formula:
66 = (1/2) * (h + 1) * h
Now, let's solve this equation step-by-step to find the values of height and base.
Step 1: Distribute the (1/2) to the terms inside the brackets:
66 = (1/2) * h^2 + (1/2) * h
Step 2: Multiply both sides of the equation by 2 to eliminate the fraction:
66 * 2 = h^2 + h
Step 3: Simplify the equation:
132 = h^2 + h
Step 4: Rewrite the equation in standard form:
h^2 + h - 132 = 0
Now, let's factorize the quadratic equation:
Step 5: Find two numbers whose sum is 1 and whose product is -132. From observation, we can find that these numbers are 11 and -12.
(h + 11)(h - 12) = 0
Step 6: Set each factor equal to zero and solve for 'h':
h + 11 = 0 or h - 12 = 0
(h + 11 = 0) gives h = -11, which is not meaningful in this context since height cannot be negative.
Thus, the valid solution is h = 12.
Now that we have found the height of the triangle as 12 meters, we can calculate the base:
base = height + 1
base = 12 + 1
base = 13
Therefore, the height of the triangle is 12 meters, and the base is 13 meters.
To find the base and height of the triangle, we can use the formula for the area of a triangle.
The formula for the area of a triangle is:
Area = 0.5 * base * height
Given that the area of the triangle is 66 m², we can set up the equation:
66 = 0.5 * base * height
Let's use the given information to find the relationship between the base and the height:
We are told that the base of the triangle is 1 meter more than the height. We can express this relationship as:
base = height + 1
Now, substitute this expression for the base in the area formula:
66 = 0.5 * (height + 1) * height
Now, let's simplify the equation:
66 = 0.5 * (height^2 + height)
Multiplying both sides of the equation by 2 to eliminate the fraction:
132 = height^2 + height
Rearranging the equation into a quadratic form:
height^2 + height - 132 = 0
This is a quadratic equation, which we can solve for the height using factoring, completing the square, or the quadratic formula. Let's use factoring to solve it.
Factoring the quadratic equation:
(height - 11)(height + 12) = 0
Setting each factor to zero and solving for the height:
height - 11 = 0 -> height = 11
height + 12 = 0 -> height = -12 (discarded since height cannot be negative in this context)
So, the height of the triangle is 11 meters.
Now, we can find the base by substituting the height value back into the relationship between the base and height:
base = height + 1 = 11 + 1 = 12
Therefore, the base of the triangle is 12 meters and the height is 11 meters.