Answer:
[tex]d=10[/tex]
Step-by-step explanation:
Pythagorean theorem:
[tex]d^2+d^2=10\sqrt{2}^2,\\2d^2=200,\\d^2=100,\\d=\boxed{10}[/tex]
Alternative:
[tex]\cos 45^{\circ}=\frac{d}{10\sqrt{2}},\\\frac{\sqrt{2}}{2}=\frac{d}{10\sqrt{2}},\\2d=10\sqrt{2}\cdot \sqrt{2},\\2d=20,\\d=\boxed{10}[/tex]
Alternative (derived from the Pythagorean theorem):
All 45-45-90 triangles have side lengths [tex]x-x-x\sqrt{2}[/tex], where [tex]x\sqrt{2}[/tex] is the hypotenuse. Since this triangle's hypotenuse is [tex]10\sqrt{2}[/tex], its sides are [tex]10-10-10\sqrt{2}[/tex].
