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.

Answer 1

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).