the base of a triangle is 5cm greater than the height. The area is 33cm^2. What is the height and the length of the base?
33=x(x+5)
find x, the height, first.
x^2+5x-33=0
use the quadratic equation
x=-5/2 +-1/2 sqrt (25+132)
= -2.5+1/2 sqrt157
ignore the negative answer.
Let's assume the height of the triangle is h cm.
According to the given information, the base of the triangle is 5 cm greater than the height. So, the length of the base would be h + 5 cm.
We know that the area of a triangle can be calculated using the formula: Area = (base * height) / 2.
Substituting the values into the formula, we have: 33 = ((h + 5) * h) / 2.
Multiplying both sides of the equation by 2 gives us: 66 = h^2 + 5h.
Rearranging the equation, we get: h^2 + 5h - 66 = 0.
To solve this quadratic equation, we can factorize it or use the quadratic formula. Let's use the quadratic formula.
The quadratic formula states: x = (-b ± √(b^2 - 4ac)) / (2a).
For our equation, a = 1, b = 5, and c = -66.
Plugging in these values, we have: h = (-5 ± √(5^2 - 4 * 1 * -66)) / (2 * 1).
Simplifying further, we get: h = (-5 ± √(25 + 264)) / 2.
This becomes: h = (-5 ± √289) / 2.
Taking the square root of 289, we get √289 = 17.
So the equation becomes: h = (-5 ± 17) / 2.
Taking the positive value, we have: h = (17 - 5) / 2, which gives us h = 6 cm.
Since the base is 5 cm greater than the height, the length of the base would be 6 + 5 = 11 cm.
Therefore, the height of the triangle is 6 cm and the length of the base is 11 cm.
To find the height and the length of the base of the triangle, we can use the formula for the area of a triangle, which is:
Area = (1/2) * base * height
Given that the area is 33 cm^2, we can write the equation as:
33 = (1/2) * base * height
Next, we're told that the base is 5 cm greater than the height. Let's represent the height as "h". Therefore, the base can be represented as "h + 5".
Now we can substitute these values into the equation:
33 = (1/2) * (h + 5) * h
To solve for the height, we need to rearrange the equation and solve for h. Let's multiply both sides of the equation by 2 to get rid of the fraction:
66 = (h + 5) * h
Expanding the equation:
66 = h^2 + 5h
Rearranging the equation to form a quadratic equation:
h^2 + 5h - 66 = 0
Now we can solve this quadratic equation for h. Factoring or using the quadratic formula will give us the two possible values for h. Solving the equation, we find two solutions: h = 6 and h = -11.
Since the height cannot be negative, we discard the solution h = -11. Therefore, the height of the triangle is 6 cm.
To find the length of the base, we substitute this value of h into our earlier expression: base = h + 5. Thus, the length of the base is 6 cm + 5 cm = 11 cm.
Therefore, the height of the triangle is 6 cm and the length of the base is 11 cm.