The area of a triangle is 66 m^2, and its base is 1 m more than the height. (See the illustration in the link.) Find the base and height of the triangle.
www.webassign.net/ewenmath10/11-1-031-alt.gif
areatriangle = (1/2) base x height
= (1/2)(x+1)(x)
= 66
(1/2)(x+1)(x) = 66
x(x+1) = 132
x^2 + x - 132 = 0
(x+12)(x-11) = 0
x = -12 or x = 11, but the sides can't be negative, so x = 11
state your conclusion
is the base is 11 and height is 10?
NO
x = 11, so what is x+1 ???
How could it be 11 and 10 ?
then the area would be (1/2)(10)(11) = 55, not 66
base would be 12 and height is 11
then the area would be (1/2)(12)(11) =66
To find the base and height of the triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Given that the area is 66 m^2, we can substitute the values and simplify the equation:
66 = (1/2) * base * height
Now we need to use the additional information provided, which states that the base is 1 meter more than the height. Let's express that in terms of one of the variables, either base or height. Let's assume the height is h. Therefore, the base would be h + 1.
Substituting these values into the equation, we get:
66 = (1/2) * (h + 1) * h
Now we solve for h by multiplying both sides by 2:
132 = (h + 1) * h
Expanding the equation:
132 = h^2 + h
Rearranging the equation in standard quadratic form:
h^2 + h - 132 = 0
Now we need to solve this quadratic equation. We can factor it or use the quadratic formula. Factoring may not be easy since the coefficients are not obvious factors of 132. Therefore, we will use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = 1, and c = -132. Substituting these values, we get:
h = (-(1) ± √((1)^2 - 4(1)(-132))) / (2(1))
Simplifying the equation:
h = (-1 ± √(1 + 528)) / 2
This becomes:
h = (-1 ± √529) / 2
Now we simplify the square root:
h = (-1 ± 23) / 2
This gives us two possible values for height: h = (-1 + 23) / 2 = 11 or h = (-1 - 23) / 2 = -12.
Since the height of a triangle cannot be negative, we disregard the negative value.
Therefore, the height (h) of the triangle is 11 meters.
To find the base, we use the information that the base is 1 meter more than the height:
base = height + 1 = 11 + 1 = 12 meters.
Therefore, the base of the triangle is 12 meters.