Final answer:
The length of side a of the isosceles triangle is √2 cm.
Step-by-step explanation:
The base of the isosceles triangle is 10cm. Let's assume that the height of the triangle is h cm. According to the given information, side a is 1cm longer than the height of the triangle, which means side a is h +1 cm. Since the triangle is isosceles, the other side b is also h cm.
Using the Pythagorean theorem, we can find the value of h. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
a^2 = b^2 + h^2
Substituting the values we have:
(h + 1)^2 = h^2 + h^2
Expanding and simplifying the equation:
h^2 + 2h + 1 = 2h^2
Simplifying further:
2h^2 - h^2 + 2h - 1 = 0
h^2 + 2h - 1 = 0
Now we can solve this quadratic equation. Let's use the quadratic formula:
h = (-2 ± √(2^2 - 4(1)(-1))) / (2(1))
h = (-2 ± √(4 + 4)) / 2
h = (-2 ± √8) / 2
h = (-2 ± 2√2) / 2
h = -1 ± √2
Since the height of the triangle cannot be negative, we take the positive value:
h = √2 - 1
Now that we have the value of h, we can find the length of side a:
a = h + 1 = (√2 - 1) + 1 = √2
Therefore, the length of side a is √2 cm.