The base of a triangle is 4 less than three times the height. The area of the triangle is 38 square cm. A=bxh divided by 2.
Let's assume the base of the triangle is x and the height is y.
Given that the base is 4 less than three times the height, we can write the equation:
x = 3y - 4
The formula for the area of a triangle is A = (1/2) * base * height. Substituting the given values, we have:
38 = (1/2) * x * y
Using the first equation, we can replace x in the second equation:
38 = (1/2) * (3y - 4) * y
Simplifying the equation:
38 = (3/2)y^2 - 2y
Rearranging the equation to set it equal to zero:
0 = (3/2)y^2 - 2y - 38
Multiplying every term by 2 to eliminate the fraction:
0 = 3y^2 - 4y - 76
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation, a = 3, b = -4, and c = -76. Plugging these values into the quadratic formula:
y = (-(-4) ± sqrt((-4)^2 - 4(3)(-76))) / (2(3))
y = (4 ± sqrt(16 + 912)) / 6
y = (4 ± sqrt(928)) / 6
y = (4 ± sqrt(16*58)) / 6
y = (4 ± 4√58) / 6
y = (2 ± 2√58) / 3
So the possible values for y are (2 + 2√58) / 3 and (2 - 2√58) / 3.
Substituting these values into the equation x = 3y - 4 to find the corresponding values for x:
x = 3((2 + 2√58) / 3) - 4
x = 2 + 2√58 - 4
x = -2 + 2√58
and
x = 3((2 - 2√58) / 3) - 4
x = 2 - 2√58 - 4
x = -2 - 2√58
Therefore, the possible values for x are -2 + 2√58 and -2 - 2√58.
So the base and height of the triangle are either (-2 + 2√58, 2 + 2√58) or (-2 - 2√58, 2 - 2√58).
To find the height and base of the triangle, we can use the given information about their relationship as well as the formula for finding the area of a triangle.
Let's first assign variables to the height and base of the triangle.
Let h be the height
Let b be the base
According to the given information, the base of the triangle is 4 less than three times the height.
So, we can write the equation:
b = 3h - 4
The formula to find the area of a triangle is:
Area = (base * height) / 2
We are also given that the area of the triangle is 38 square cm. So, we can write the equation:
38 = (b * h) / 2
Now, we have a system of two equations:
b = 3h - 4
38 = (b * h) / 2
To solve this system of equations, we can substitute the value of b from the first equation into the second equation:
38 = ((3h - 4) * h) / 2
Multiplying both sides of the equation by 2 to eliminate the fraction gives:
76 = (3h - 4) * h
Expanding the right side of the equation:
76 = 3h^2 - 4h
Bringing all the terms to one side:
3h^2 - 4h - 76 = 0
Now we have a quadratic equation, which we can solve to find the value(s) of h.
By factoring, completing the square, or using the quadratic formula, we can find that h = 8 or h = -3.
Since the height of a triangle cannot be negative, the height of the triangle is 8 cm.
Now we can substitute this value of h back into the first equation to find the base:
b = 3h - 4
b = 3(8) - 4
b = 24 - 4
b = 20
Therefore, the height of the triangle is 8 cm and the base is 20 cm.
To solve this problem, we need to set up an equation using the given information. Let's use "b" to represent the base and "h" to represent the height.
According to the problem, the base of the triangle is 4 less than three times the height. This can be written as:
b = 3h - 4
The formula to calculate the area of a triangle is given by:
A = (b * h) / 2
We are also given that the area of the triangle is 38 square cm. So we can write:
38 = (b * h) / 2
Now we have two equations:
1) b = 3h - 4
2) 38 = (b * h) / 2
To solve this system of equations, we can substitute equation 1 into equation 2:
38 = ((3h - 4) * h) / 2
Multiplying both sides of the equation by 2 to remove the denominator:
76 = (3h - 4) * h
Expanding the equation:
76 = 3h^2 - 4h
Rearranging the equation:
3h^2 - 4h - 76 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Since the quadratic equation doesn't factor nicely, let's use the quadratic formula:
h = (-b ± √(b^2 - 4ac)) / 2a
In this equation, a = 3, b = -4, and c = -76. Substituting these values:
h = (-(-4) ± √((-4)^2 - 4 * 3 * -76)) / (2 * 3)
h = (4 ± √(16 + 912)) / 6
h = (4 ± √928) / 6
Simplifying the square root:
h = (4 ± √(16 * 58)) / 6
h = (4 ± 4√58) / 6
h = (2 ± 2√58) / 3
From here, we have two possible values for the height. Let's calculate the corresponding base for each value of the height using equation 1:
For h = (2 + 2√58) / 3:
b = 3h - 4
b = 3((2 + 2√58) / 3) - 4
b = 2 + 2√58 - 4
b = -2 + 2√58
For h = (2 - 2√58) / 3:
b = 3h - 4
b = 3((2 - 2√58) / 3) - 4
b = 2 - 2√58 - 4
b = -2 - 2√58
So the two possible sets of height and base values for the triangle are:
1) h = (2 + 2√58) / 3, b = -2 + 2√58
2) h = (2 - 2√58) / 3, b = -2 - 2√58