The base of a triangle is four less then three times the height the area of the triangle is 38 square centimeters
Let's call the height of the triangle h and the base of the triangle b.
According to the problem, the base of the triangle is 4 less than three times the height, so we can write an equation for this:
b = 3h - 4
The area of a triangle is calculated as (1/2) * base * height, so we can use this formula with the given area of 38 square centimeters:
(1/2) * b * h = 38
Substituting the value of b from the first equation into the second equation:
(1/2) * (3h - 4) * h = 38
Expanding the equation, we get:
(3h^2 - 4h) / 2 = 38
Multiplying both sides of the equation by 2:
3h^2 - 4h = 76
Rearranging the equation to make it quadratic:
3h^2 - 4h - 76 = 0
Now we can solve this quadratic equation to find the value of h. We can either factor it or use the quadratic formula:
Using factoring:
(3h + 14)(h - 4) = 0
Setting each factor equal to zero:
3h + 14 = 0 or h - 4 = 0
Simplifying each equation:
3h = -14 or h = 4
Solving for h in each equation:
h = -14/3 or h = 4
Since the height of a triangle cannot be negative, we can disregard the solution h = -14/3.
Therefore, the height of the triangle is 4 centimeters.
Substituting this value into the equation for b:
b = 3h - 4
b = 3(4) - 4
b = 12 - 4
b = 8
So, the base of the triangle is 8 centimeters.
In conclusion, the height of the triangle is 4 cm and the base of the triangle is 8 cm.
To find the base and height of the triangle, we can set up an equation using the given information.
Let's denote the height of the triangle as "h" and the base as "b".
The first piece of information tells us that the base is four less than three times the height:
b = 3h - 4
The second piece of information tells us that the area of the triangle is 38 square centimeters:
(1/2)*b*h = 38
Now we have a system of two equations:
b = 3h - 4
(1/2)*b*h = 38
We can solve this system of equations by substituting the value of b from the first equation into the second equation.
(1/2)*(3h - 4)*h = 38
Simplifying:
(3h - 4)*h = 76
Expanding:
3h^2 - 4h = 76
Rearranging the equation and setting it equal to 0:
3h^2 - 4h - 76 = 0
Now we can solve this quadratic equation. We can either factor it or use the quadratic formula, let's use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 3, b = -4, and c = -76. Plugging in these values:
h = (-(-4) ± √((-4)^2 - 4(3)(-76))) / (2*3)
= (4 ± √(16 + 912)) / 6
= (4 ± √928) / 6
Now we find the square root of 928:
√928 ≈ 30.463
Therefore, h = (4 ± 30.463) / 6
Now we can find two possible values for h:
h1 = (4 + 30.463) / 6 ≈ 5.744
h2 = (4 - 30.463) / 6 ≈ -4.077
Since the height of a triangle cannot be negative, we discard the second solution.
Now we substitute the value of h = 5.744 into the first equation to find the value of b:
b = 3h - 4
= 3(5.744) - 4
≈ 13.232
Therefore, the base of the triangle is approximately 13.232 centimeters and the height is approximately 5.744 centimeters.