One of the legs of a right-angled triangle is 2cm larger than the other, and the area is less than 60cm2.

One of the legs of a right-angled triangle is 2cm larger than the other, and the area is less than 60cm2. How long are the legs?

1. Let a and b be the legs of a right-angled triangle, with a <b. Then, according to the problem statement:
b = a + 2. (1)
2. The area of ​​a right-angled triangle is half the product of the legs, therefore:
1/2 * ab <60. (2)
3. Let us compose the system from equation (1) and inequality (2):
{b = a + 2;
{1/2 * ab <60;
{b = a + 2;
{ab <2 * 60;
{b = a + 2;
{a (a + 2) <120;
{b = a + 2;
{a ^ 2 + 2a – 120 <0.
4. Find the roots of the square trinomial and solve the inequality:
D / 4 = 1 ^ 2 + 120 = 121;
a = -1 ± √121 = -1 ± 11;
a1 = -1 – 11 = -12;
a2 = -1 + 11 = 10;
a ∈ (-12; 10).
5. Back to the system:
{b = a + 2;
{a ∈ (-12; 10).
In the sense of the problem, a> 0, so let’s make the constraint:
{b = a + 2;
{a ∈ (0; 10);
{b ∈ (2; 12);
{a ∈ (0; 10).
Answer: b ∈ (2; 12).