The base of a triangle is 8 cm greater than the height. The area is 24cm^2. What is the base and height?
jjik
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. So, the base can be represented as: 'h + 8' cm.
The formula for calculating the area of a triangle is:
Area = (1/2) * base * height
Given that the area is 24 cm^2, we can now form the equation:
24 = (1/2) * (h + 8) * h
Now, let's solve the equation step-by-step:
Step 1: Distribute:
24 = (1/2)(h^2 + 8h)
Step 2: Expand the brackets:
24 = (1/2)h^2 + 4h
Step 3: Multiply each term by 2 to eliminate the fraction:
48 = h^2 + 8h
Step 4: Rearrange the equation in quadratic form (ax^2 + bx + c = 0):
h^2 + 8h - 48 = 0
Step 5: Factorize the quadratic equation:
(h + 12)(h - 4) = 0
Step 6: Solve for h:
h + 12 = 0 or h - 4 = 0
So, we have two possible solutions for height:
h = -12 or h = 4
However, since a negative height does not make sense in this context, we can discard h = -12.
Therefore, the height of the triangle is h = 4 cm.
To find the base, we can substitute this value back into the equation: base = h + 8
base = 4 + 8
base = 12 cm
So, the base of the triangle is 12 cm and the height is 4 cm.
To find the base and height of a triangle given its area, we need to use the formula for the area of a triangle:
Area = (base * height) / 2
Let's denote the height of the triangle as 'h' cm. According to the given information, the base is 8 cm greater than the height, so the base can be represented as 'h+8' cm.
Now we can substitute these values into the formula for the area:
24 = ((h+8) * h) / 2
To solve this equation, we will multiply both sides of the equation by 2 to eliminate the fraction:
48 = (h+8) * h
Expanding the equation, we get:
48 = h^2 + 8h
Rearranging the terms to form a quadratic equation in standard form:
h^2 + 8h - 48 = 0
Now, we can solve this quadratic equation either by factoring or by using the quadratic formula. Since factoring might be a bit tricky in this case, let's use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / (2a)
For the equation h^2 + 8h - 48 = 0, we have a = 1, b = 8, and c = -48. Substituting these values into the formula, we get:
h = (-8 ± √(8^2 - 4*1*(-48))) / (2*1)
Simplifying further:
h = (-8 ± √(64 + 192)) / 2
h = (-8 ± √(256)) / 2
Taking the square root of 256:
h = (-8 ± 16) / 2
Now we can compute the two possible values for the height (h):
1. h = (-8 + 16) / 2
h = 8 / 2
h = 4 cm
2. h = (-8 - 16) / 2
h = -24 / 2
h = -12 cm
Since the dimensions of a triangle cannot be negative, we discard the negative value and conclude that the height of the triangle is 4 cm.
Now, we can find the base by substituting the value of 'h' into the expression 'h+8':
Base = 4 + 8
Base = 12 cm
Therefore, the height of the triangle is 4 cm and the base is 12 cm.
Let height is x.
So base=x+8
area of triangle=1/2*b*h
given area=24
so 24=1/2*b*h
2*24=(x+8)*x
48=(x^2+8x)
x^2+8x-48=0
x^2+(12x-4x)-48=0
x^2+12x-4x-48=0
x(x+12)-4(x+12)=0
(x+12)(x-4)=0
x=4,-12
so taking x=4
height=4cm
base=x+8=4+8=12cm