the height of a triangle is 4cm shorter than its base. if the triangle has an area of 48cm2, find the length of the base of the triangle.
Let h=height
base, b =h+4
Area of triangle
=bh/2
=(h+4)h/2
or
h(h+4)/2 = 48
Solve for h.
h²+4h-96=0
(h-12)(h+8)=0
h=12 or h=-8 (reject negative root)
To find the length of the base of the triangle, we can use the formula for the area of a triangle: A = 1/2 * base * height.
Let's start by assigning variables to the given information:
Let b be the length of the base of the triangle.
The height of the triangle is 4cm shorter than its base, so the height would be (b - 4) cm.
Now, we know that the area of the triangle is 48cm^2. So we can set up the equation:
48 = 1/2 * b * (b - 4)
To solve this equation, we can simplify it step by step:
Step 1: Distribute 1/2 to the terms inside the parentheses:
48 = 1/2 * (b^2 - 4b)
Step 2: Simplify the equation further:
48 = (1/2) * b^2 - (1/2) * 4b
Step 3: Simplify the equation even more:
48 = (1/2) * b^2 - 2b
Step 4: Multiply both sides of the equation by 2 to eliminate the fraction:
96 = b^2 - 4b
Step 5: Rearrange the equation to get a quadratic equation:
b^2 - 4b - 96 = 0
Step 6: Solve the quadratic equation. Factoring won't work in this case, so we can use the quadratic formula:
b = (-(-4) ± √((-4)^2 - 4*1*(-96))) / (2*1)
Simplifying further:
b = (4 ± √(16 + 384)) / 2
b = (4 ± √400) / 2
b = (4 ± 20) / 2
Now we have two possible solutions:
b1 = (4 + 20) / 2 = 24 / 2 = 12
b2 = (4 - 20) / 2 = -16 / 2 = -8
Since the base length cannot be negative, the only valid solution is b = 12.
Therefore, the length of the base of the triangle is 12 cm.