Answer:
The longer leg measures 4 + √2.
Explanation:
Let a = length of the short leg.
Let b = length of the long leg.
Let c = length of the hypotenuse.
area = ab/2
perimeter = a + b + c
Pythagorean theorem: a² + b² = c²
ab/2 = 7
a + b + c = 14
a² + b² = c²
ab = 14
a + b + c = 14
a² + b² = c²
b = 14/a
c = 14 - a - b
a² + b² = c²
b = 14/a
c = 14 - a - b
a² + 196/a² = (14 - a - b)²
a² + 196/a² = (14 - a - 14/a)²
a² + 196/a² = 196 - 14a - 196/a - 14a + a² + 14 - 196/a + 14 + 196/a²
0 = 196 - 14a - 196/a - 14a + 14 - 196/a + 14
0 = 196a - 14a² - 196 - 14a² + 14a - 196 + 14a
28a² - 224a + 392 = 0
a² - 8a + 14 = 0
a² - 8a + 16 = -14 + 16
(a - 4)² = 2
a - 4 = ±√2
a = 4 ± √2
Let a = 4 - √2 (short leg)
ab/2 = 7
(4 - √2)b/2 = 7
(4 - √2)b = 14
b = 14/(4 - √2)
b = 14/(4 - √2) × (4 + √2/(4 + √2)
b = 14(4 + √2)/(16 - 2)
b = 14(4 + √2)/14
b = 4 + √2 (long leg)
a + b + c = 14
4 + √2 + 4 - √2 + c = 14
8 + c = 14
c = 6 (hypotenuse)
Answer: The longer leg measures 4 + √2.