the height of a triangle is 5 cm more than the length of the base. if the area of a triangle is 133 cm^2 find the hight nad length of the base

To find the height and length of the base of the triangle, we need to solve two equations involving the height and base of the triangle.

Let's assign variables to the height and base of the triangle:

Let h = height of the triangle
Let b = length of the base of the triangle

According to the problem, the height of the triangle is 5 cm more than the length of the base. So we can write the first equation as:

h = b + 5 -------- (Equation 1)

The area of a triangle is given by the formula:

Area = (1/2) * base * height

In this case, the area is given as 133 cm^2. Substituting the given values into the formula, we get:

133 = (1/2) * b * h ------- (Equation 2)

Now we have two equations with two variables (Equation 1 and Equation 2). We can solve them simultaneously to find the values of h and b.

Let's substitute the value of h from Equation 1 into Equation 2:

133 = (1/2) * b * (b + 5)

Expanding the equation:

133 = (1/2) * (b^2 + 5b)

Multiplying both sides by 2 to remove the fraction:

266 = b^2 + 5b

Rearranging the equation to form a quadratic equation:

b^2 + 5b - 266 = 0

Now we can solve this quadratic equation to find the length of the base, b. We can factorize it or use the quadratic formula to find the solutions.

Using the quadratic formula:

b = (-5 ± √(5^2 - 4 * 1 * -266)) / (2 * 1)

Simplifying further:

b = (-5 ± √(25 + 1064)) / 2
b = (-5 ± √1089) / 2

Taking the positive value:

b = (-5 + 33) / 2 or b = (-5 - 33) / 2

b = 28 / 2 or b = -38 / 2
b = 14 or b = -19

Since the length cannot be negative, we discard the negative value.

So, the length of the base, b, is 14 cm.

Now, substitute this value of b into Equation 1 to find the height:

h = 14 + 5
h = 19

The height of the triangle is 19 cm and the length of the base is 14 cm.

a = bh/2

133 = b(b+5)/2
b(b+5) = 266
b=14, so h=19