The base of a triangle is 8 cm greater than the height. The area is 64 cm^2. Find the height and the length of the base.
height = X-cm,
Base = (X + 8)cm,
A = bh/2 = (X + 8)X/2 = 64,
Multiply both sides by 2:
(X + 8)X = 128,
X^2 + 8X = 128,
X^2 + 8X - 128 = 0,
(X - 8)(X + 16) = 0,
X - 8 = 0,
X = 8.
X + 16 = 0,
X = -16.
Use the + value of X:
X = 8cm = height.
X + 8 = 8 + 8 = 16cm = base.
Well, well, well, looks like we have a triangle in need of some dimensions! Let's solve this puzzle, shall we?
Let's call our height 'h' because it likes to stay humble. And since the base is 8 cm greater than the height, we'll call it 'h + 8'. Are you following along?
Now, the area of a triangle is calculated using the formula A = (base * height) / 2. We know the area is 64 cm², so we can set up our equation like this:
64 = ((h + 8) * h) / 2
Are you still with me? Great! Let's simplify this expression by getting rid of that pesky division of 2. Multiply both sides by 2, and we have:
128 = (h + 8) * h
Now, let's expand this equation:
128 = h² + 8h
Ooh la la, we've got ourselves a quadratic equation! Let's bring all the terms to one side:
h² + 8h - 128 = 0
Now, I'll just whip out my trusty quadratic solver...
*Dramatic pause*
Haha! Just clowning around there. Let's solve this equation using factoring or the quadratic formula. After some number crunching, we find that h = 8 and h = -16.
Since we're dealing with lengths, we'll go ahead and discard that negative solution. So, the height is 8 cm. And since the base is 8 cm greater, the base length is 16 cm.
And there you have it! The height of the triangle is 8 cm, and the length of the base is 16 cm. Triangle mystery solved!
Let's assume the height of the triangle is 'h' cm.
According to the given information, the base of the triangle is 8 cm greater than the height. Therefore, the length of the base is 'h + 8' cm.
The formula to calculate the area of a triangle is: A = (1/2) * base * height
Using this formula, we can determine the height and length of the base.
Given that the area of the triangle is 64 cm^2, we can write the equation as:
64 = (1/2) * (h + 8) * h
Now, let's solve this equation step-by-step to find the height (h):
Step 1: Expand the equation
64 = (1/2) * (h^2 + 8h)
Step 2: Distribute (1/2) to both terms inside the parentheses
64 = (1/2) * h^2 + (1/2) * 8h
Step 3: Simplify the equation
64 = (1/2)h^2 + 4h
Step 4: Multiply both sides of the equation by 2 to eliminate the fraction
128 = h^2 + 8h
Step 5: Rearrange the equation to make it equal to zero
h^2 + 8h - 128 = 0
Step 6: Factorize the quadratic equation (if possible)
(h + 16)(h - 8) = 0
Step 7: Set each factor equal to zero and solve for 'h'
h + 16 = 0 or h - 8 = 0
h = -16 or h = 8
Since the height cannot be negative, the height of the triangle is 8 cm.
Now, substitute this value back into the equation for the base to find its length:
base = height + 8 = 8 + 8 = 16 cm
Therefore, the height of the triangle is 8 cm, and the length of the base is 16 cm.
To find the height and the length of the base of the triangle, we can use the formula to calculate the area of a triangle.
Step 1: Let's assume the height of the triangle as 'h' cm.
Step 2: According to the problem, the base of the triangle is 8 cm greater than the height. So the length of the base is (h + 8) cm.
Step 3: The formula to calculate the area of a triangle is (1/2) * base * height. In this case, the area is given as 64 cm^2. So we have: (1/2) * (h + 8) * h = 64.
Step 4: Simplify the equation: (0.5h^2 + 4h) = 64.
Step 5: Multiply both sides of the equation by 2 to eliminate the fraction: h^2 + 8h = 128.
Step 6: Rewrite the equation in the standard form: h^2 + 8h - 128 = 0.
Step 7: Solve the quadratic equation to find the value of h. In this case, h = 8 cm or h = -16 cm.
However, since height cannot be negative, we discard the value h = -16 cm.
Therefore, the height of the triangle is h = 8 cm.
To find the length of the base, substitute the value of h into the expression (h + 8): base = 8 + 8 = 16 cm.
So, the height of the triangle is 8 cm and the length of the base is 16 cm.