The hypotenuse of a right triangle is 5cm long. One leg is 3cm longer than the other leg. Find the dimensions of the shorter leg. (give your answer in radical form)

This is an application of Pythagoras' Theorem

x^ + (x+3)^2 = 5^2

expand and solve using the quadratic equation formula.

To solve this problem, let's assume that the shorter leg of the right triangle is x cm.

According to the given information, the other leg (which is 3 cm longer) can be represented as (x + 3) cm.

Now, we can use the Pythagorean theorem to relate the lengths of the triangle's sides. The theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

Using this information, we can write the equation as follows:

x^2 + (x + 3)^2 = 5^2

Expanding the equation, we get:

x^2 + (x + 3)(x + 3) = 25
x^2 + x^2 + 6x + 9 = 25

Combining like terms, we have:

2x^2 + 6x + 9 = 25

Rearranging the equation, we get:

2x^2 + 6x + 9 - 25 = 0
2x^2 + 6x - 16 = 0

We can now solve the quadratic equation to find the value of x. To do this, we can either factorize, complete the square, or use the quadratic formula. In this case, we will use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

For our equation of 2x^2 + 6x - 16 = 0, a = 2, b = 6, and c = -16. Plugging these values into the quadratic formula, we get:

x = (-6 ± √(6^2 - 4 * 2 * -16)) / (2 * 2)
x = (-6 ± √(36 + 128)) / 4
x = (-6 ± √164) / 4

Simplifying further, we have:

x = (-6 ± √(4 * 41)) / 4
x = (-6 ± 2√41) / 4

Now, we can simplify the expression by factoring out the common factor 2 in the numerator and denominator:

x = 2(-3 ± √41) / 4

Simplifying the fraction by canceling out 2 and 4, we get:

x = -3 ± √41/2

Therefore, the dimensions of the shorter leg of the right triangle are approximately:

x ≈ -3 + √41/2
and
x ≈ -3 - √41/2

Since the length of a side cannot be negative in this context, we can discard the negative solution.

Thus, the dimension of the shorter leg (in radical form) is:

x ≈ -3 + √41/2