ok supposed the base of a triangle is 5 cm greater than the height the area is 52 cm what is the height and length at the base?

well if it's an equilateral triangle

h=9.49035
b=10.95851
and if it's a right triangle
h=8
b=13

Do you know what type of triangle this is?

Let's denote the height of the triangle as "h" and the length at the base as "b".

According to the given information, the base is 5 cm greater than the height:
b = h + 5

The formula to calculate the area of a triangle is given by:
Area = (1/2) * base * height

Substituting the given values, we have:
52 = (1/2) * b * h

Rearranging the equation in terms of "b" and substituting the relationship between b and h, we get:
52 = (1/2) * (h + 5) * h

Expanding the equation, we have:
104 = h^2 + 5h

Rearranging, we get the equation in standard form:
h^2 + 5h - 104 = 0

We can solve this quadratic equation to find the values of "h" and then substitute it back to find the value of "b".

To find the height and base length of a triangle given the area and a relation between them, we can solve the problem using algebraic equations.

Let's denote the height of the triangle as 'h' and the base length as 'b'. According to the problem, the base length is 5 cm greater than the height, so we can write a equation for the base length:

b = h + 5

The formula for the area of a triangle is:

Area = (1/2) * base * height

Given that the area is 52 cm², we can substitute the values into the formula:

52 = (1/2) * b * h

Since we have two equations, we can solve them simultaneously to find the values of height (h) and base length (b).

Let's substitute the value of 'b' from the first equation into the second equation:

52 = (1/2) * (h + 5) * h

Now, we can simplify the equation:

52 = (1/2) * (h² + 5h)
104 = h² + 5h

Rearranging the equation to a quadratic form:

h² + 5h - 104 = 0

To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:

h = (-5 ± √(5² - 4 * 1 * -104)) / (2 * 1)
h = (-5 ± √(25 + 416)) / 2
h = (-5 ± √441) / 2

Taking the positive value for h, we get:

h = (-5 + √441) / 2
h = (-5 + 21) / 2
h = 16 / 2
h = 8

So, the height of the triangle is 8 cm.

Now, substitute the value of h into the first equation to find the base length:

b = h + 5
b = 8 + 5
b = 13

Therefore, the height of the triangle is 8 cm and the length of the base is 13 cm.