1. Let’s assume that x=5 and y=5.

2. The Pythagorean theorem states that the hypotenuse =

3. We can use the Pythagorean theorem to show that the hypotenuse would be equal to

4. Now if we increase x to 6, then y must go down to 4, and the hypotenuse would be equal to

5. We can see from the table below that the minimum would be at x=5 and y=5. Any change would increase the length of the hypotenuse.

x	y	h
1.00	9.00	9.055
1.10	8.90	8.968
1.20	8.80	8.881
1.30	8.70	8.797
1.40	8.60	8.713
1.50	8.50	8.631
1.60	8.40	8.551
1.70	8.30	8.472
1.80	8.20	8.395
1.90	8.10	8.320
2.00	8.00	8.246
2.10	7.90	8.174
2.20	7.80	8.104
2.30	7.70	8.036
2.40	7.60	7.970
2.50	7.50	7.906
2.60	7.40	7.843
2.70	7.30	7.783
2.80	7.20	7.725
2.90	7.10	7.669
3.00	7.00	7.616
3.10	6.90	7.564
3.20	6.80	7.515
3.30	6.70	7.469
3.40	6.60	7.424
3.50	6.50	7.382
3.60	6.40	7.343
3.70	6.30	7.306
3.80	6.20	7.272
3.90	6.10	7.240
4.00	6.00	7.211
4.10	5.90	7.185
4.20	5.80	7.161
4.30	5.70	7.140
4.40	5.60	7.122
4.50	5.50	7.106
4.60	5.40	7.094
4.70	5.30	7.084
4.80	5.20	7.077
4.90	5.10	7.072
5.00	5.00	7.071
5.10	4.90	7.072
5.20	4.80	7.077
5.30	4.70	7.084
5.40	4.60	7.094
5.50	4.50	7.106
5.60	4.40	7.122
5.70	4.30	7.140
5.80	4.20	7.161
5.90	4.10	7.185
6.00	4.00	7.211
6.10	3.90	7.240
6.20	3.80	7.272
6.30	3.70	7.306
6.40	3.60	7.343
6.50	3.50	7.382
6.60	3.40	7.424
6.70	3.30	7.469
6.80	3.20	7.515
6.90	3.10	7.564
7.00	3.00	7.616
7.10	2.90	7.669
7.20	2.80	7.725
7.30	2.70	7.783
7.40	2.60	7.843
7.50	2.50	7.906
7.60	2.40	7.970
7.70	2.30	8.036
7.80	2.20	8.104
7.90	2.10	8.174
8.00	2.00	8.246
8.10	1.90	8.320
8.20	1.80	8.395
8.30	1.70	8.472
8.40	1.60	8.551
8.50	1.50	8.631
8.60	1.40	8.713
8.70	1.30	8.797
8.80	1.20	8.881
8.90	1.10	8.968
9.00	1.00	9.055
9.10	0.90	9.144
9.20	0.80	9.235
9.30	0.70	9.326
9.40	0.60	9.419
9.50	0.50	9.513
9.60	0.40	9.608
9.70	0.30	9.705
9.80	0.20	9.802
9.90	0.10	9.901
10.00	0.00	10.000