Given:

b = 169

To find:

a, c, d

Solution:

We need to solve the given equation to find the values of a, c, and d.


a² + c² = b² - d²

Substituting the value of b,

a² + c² = 169 - d²

We need to make some assumptions to find the values of a, c, and d. Let's assume that a, c, and d are integers.

Finding d:

As d is an integer, we can assume that d is less than or equal to 13 because d² > 169 - d² will not hold true for any value of d greater than 13.

Let's try different values of d to find the value that satisfies the equation.

When d = 1,
169 - d² = 168
a² + c² = 168
As a and c are integers, this equation does not have any solutions.

When d = 2,
169 - d² = 165
a² + c² = 165
This equation also does not have any solutions.

When d = 3,
169 - d² = 160
a² + c² = 160
This equation also does not have any solutions.

When d = 4,
169 - d² = 155
a² + c² = 155
This equation also does not have any solutions.

When d = 5,
169 - d² = 150
a² + c² = 150
This equation also does not have any solutions.

When d = 6,
169 - d² = 143
a² + c² = 143
This equation also does not have any solutions.

When d = 7,
169 - d² = 138
a² + c² = 138
This equation also does not have any solutions.

When d = 8,
169 - d² = 133
a² + c² = 133
This equation also does not have any solutions.

When d = 9,
169 - d² = 120
a² + c² = 120
This equation has solutions, a = 6 and c = 6.

When d = 10,
169 - d² = 119
a² + c² = 119
This equation also does not have any solutions.

When d = 11,
169 - d² = 118
a² + c² = 118
This equation also does not have any solutions.

When d = 12,
169 - d² = 145
a² + c² = 145
This equation also does not have any solutions.

When d = 13,
169 - d² = 140
a² + c² = 140
This equation also does not have any solutions.

Therefore, the only solution for a² + c² = 169 - d² is a = 6, c = 6, and d = 9.

Final Answer:

a = 6

c = 6

d = 9