the altitude of a triagnle is 2cm shorter then its base. the area is 15cm squared. determine the base of the triangle.
Why did the triangle feel a little down? Because it had a short altitude! Don't worry, I'll cheer it up and help you find the base.
Let's use a little math to solve this. We know that the area of a triangle is given by the formula: Area = (base * altitude) / 2.
So, if the altitude is 2cm shorter than the base, we can say that the altitude is (base - 2) cm.
Now, let's substitute these values into the formula: 15 = (base * (base - 2)) / 2.
To make it a bit easier, let's get rid of that fraction by multiplying both sides by 2: 30 = base * (base - 2).
Expanding this equation, we get: 30 = base^2 - 2base.
Rearranging things, we have a quadratic equation: base^2 - 2base - 30 = 0.
Now, we can solve this quadratic equation using factoring or the quadratic formula to find the value of the base.
To find the base of the triangle, we can use the formula for the area of a triangle, which is given by:
Area = (1/2) * base * altitude
Given that the altitude is 2 cm shorter than the base and the area is 15 cm², we can set up the following equation:
15 = (1/2) * base * (base - 2)
Let's solve this equation step-by-step:
Step 1: Distribute the (1/2) through the expression (base - 2):
15 = (1/2) * (base^2 - 2base)
Step 2: Multiply both sides of the equation by 2 to eliminate the fraction:
30 = base^2 - 2base
Step 3: Rearrange the equation by moving all terms to one side:
base^2 - 2base - 30 = 0
Step 4: Factorize the quadratic equation:
(base - 6)(base + 5) = 0
Step 5: Solve for base by setting each factor equal to zero:
base - 6 = 0 or base + 5 = 0
If base - 6 = 0, then base = 6.
If base + 5 = 0, then base = -5.
Since the base of a triangle cannot be negative, we discard the extraneous solution. Therefore, the base of the triangle is 6 cm.
To determine the base of the triangle, we can use the formula for the area of a triangle: A = (1/2) * base * height.
Let's represent the base of the triangle as "b" and the height as "h". According to the given information, the height is 2 cm shorter than the base. So, we have:
h = b - 2
Now, substituting the value of height in terms of the base in the area formula:
15 = (1/2) * b * (b - 2)
To solve for the base, we need to simplify and solve the quadratic equation:
15 = (1/2) * b^2 - b
Multiplying both sides by 2 to eliminate the fraction:
30 = b^2 - 2b
Rearranging the equation:
b^2 - 2b - 30 = 0
Now, we can factor the quadratic equation or use the quadratic formula to find the values of "b". In this case, let's factor the equation:
(b - 6)(b + 5) = 0
Setting each factor to zero:
b - 6 = 0 or b + 5 = 0
Solving these equations, we find:
b = 6 or b = -5
Since the base of a triangle cannot be negative, the base of the triangle is 6 cm.
Assuming "15cm squared." means 15 cm² (and not (15 cm)²).
Let
b=base of triangle
b(b-2)/2 = 15
solve for b (it's a quadratic equation).