Final answer:
The perimeter of the isosceles right triangle with a hypotenuse of length 8 is approximately 19.31 units. The area of the triangle is 16 square units.
Step-by-step explanation:
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). For an isosceles right triangle, the lengths of the two legs (a and b) are equal. Thus, when the hypotenuse (c) is 8, we can express this relationship as:
a2 + a2 = 82
2a2 = 64
a2 = 32
a = √32
a = 4√2
To find the perimeter of the triangle, we add the lengths of all three sides:
Perimeter = a + a + c
Perimeter = 4√2 + 4√2 + 8
Perimeter = 8√2 + 8, which is approximately 19.31
To find the area of the triangle:
Area = 1/2 × base × height
Since the triangle is isosceles and right-angled, the base and height are the same, both equal to a.
Area = 1/2 × 4√2 × 4√2
Area = 1/2 × 32, which is 16 square units.