Here is a right- angled triangle.
|\
| \
x | \
|___\
x - 2
All measurements are in centimetres.
The area if the triangle is 2.5cm^2.
Find the perimeter of the triangle.
Key values:
Equation for area:
?
- __(___) = ?
?
Quadratic equation:
-b ± √b^2 - 4ac
-------------
2a
Solutions:
x = ? or ?
Length of the hypotenuse:
?
Perimeter
? + ? + ? = ? cm
Thanks
looks like your base is x-2 and the height is x, so
(1/2)(x)(x-2) = 2.5
x(x-2) = 5
x^2 - 2x - 5 = 0
solve for x and take it from there
Hi Reiny,
What would a, b and c be for the quadratic formula?
Thanks
huh? a,b,c are the coefficients in the equation, as usual!
Edhbshan
To find the perimeter of the triangle, we need to determine the lengths of all three sides.
Let's denote the length of the side opposite the right angle as x and the length of the other side as x-2. Since the area of the triangle is given as 2.5 cm^2, we can use the formula for the area of a triangle to set up an equation:
(area of triangle) = (base × height) / 2
Substituting the given values, we have:
2.5 = (x × (x-2)) / 2
Now, let's simplify the equation and solve it.
2.5 = (x^2 - 2x) / 2
Multiplying both sides by 2 to eliminate the fraction:
5 = x^2 - 2x
Rearranging the equation:
x^2 - 2x - 5 = 0
This is a quadratic equation in standard form, where:
a = 1
b = -2
c = -5
Using the quadratic formula, we can find the solutions for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (-(-2) ± √((-2)^2 - 4(1)(-5))) / (2(1))
x = (2 ± √(4 + 20)) / 2
x = (2 ± √24) / 2
x = (2 ± √(4 × 6)) / 2
x = (2 ± 2√6) / 2
x = 1 ± √6
So, the possible values for x are 1 + √6 and 1 - √6.
Next, let's find the length of the hypotenuse using the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Using the given values:
Length of the hypotenuse squared = x^2 + (x-2)^2
Substituting the possible values for x:
Length of the hypotenuse squared = (1 + √6)^2 + ((1 + √6) - 2)^2
= (1 + √6)^2 + (-1 + √6)^2
Expanding and simplifying:
Length of the hypotenuse squared = 1 + 2√6 + 6 + 1 + 2√6 + 6
= 9 + 4√6
Taking the square root of both sides:
Length of the hypotenuse = √(9 + 4√6)
Finally, we can calculate the perimeter of the triangle by summing all three sides:
Perimeter = x + (x-2) + hypotenuse
Substituting the values for x and the length of the hypotenuse:
Perimeter = (1 + √6) + ((1 + √6) - 2) + √(9 + 4√6)
Now, simplify this expression to find the perimeter of the triangle.
I hope this explanation helps you understand the process of finding the perimeter of the triangle given the area and the given measurements.