In this right triangle, you are given the measurements for the hypotenuse, c, and one leg, b. The hypotenuse is always opposite the right angle and it is always the longest side of the triangle. To find the length of leg a, substitute the known values into the Pythagorean Theorem. Solve for a2.

Given:
We are given a right-angled triangle and we need to find the lengths of the unknown sides.

Triangle 1:
In the first right-angled triangle, we are given the lengths of two sides.

Given sides:
- One side is 5 cm long.
- Another side is 12 cm long.

To find:
We need to find the length of the third side, which is the hypotenuse.

Solution:
We can use the Pythagorean theorem to find the length of the unknown side.

Pythagorean theorem:
In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

Mathematical expression:
c^2 = a^2 + b^2

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

Substituting the given values:
c^2 = 5^2 + 12^2
c^2 = 25 + 144
c^2 = 169

Taking the square root of both sides:
c = √169
c = 13 cm

Therefore, the length of the unknown side in triangle 1 is 13 cm.

Triangle 2:
In the second right-angled triangle, we are given the lengths of one side and the hypotenuse.

Given sides:
- One side is 8 cm long.
- The hypotenuse is 17 cm long.

To find:
We need to find the length of the third side.

Solution:
Again, we can use the Pythagorean theorem to find the length of the unknown side.

Pythagorean theorem:
c^2 = a^2 + b^2

Mathematical expression:
17^2 = 8^2 + b^2

Substituting the given values:
289 = 64 + b^2
b^2 = 289 - 64
b^2 = 225

Taking the square root of both sides:
b = √225
b = 15 cm

Therefore, the length of the unknown side in triangle 2 is 15 cm.

Summary:
- In the first triangle, the length of the unknown side is 13 cm.
- In the second triangle, the length of the unknown side is 15 cm.

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