66. 2) Find a and b. A a=10surd 2,b=10surd 2 A B a=5surd 2,b=5surd 2 B a=10,b=10surd 2 D a=5,b=5

To find the lengths of the legs \(a\) and \(b\) of the right triangle, we can use the information given in the picture details.Step 1: Use the Pythagorean theorem.The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is:\[c^2 = a^2 + b^2\]Step 2: Use the given angle to determine the relationship between the legs.Since one of the angles is 45° and we are dealing with a right triangle, the other non-right angle must also be 45° (since the angles in a triangle add up to 180° and we already have one 90° angle). This means the triangle is an isosceles right triangle, and the legs opposite the 45° angles are equal in length. Therefore, \(a = b\).Step 3: Apply the information to the Pythagorean theorem.Since \(a = b\) and the hypotenuse \(c = 10\), we can write:\[10^2 = a^2 + a^2\]\[100 = 2a^2\]\[50 = a^2\]Step 4: Solve for \(a\).To find \(a\), we take the square root of both sides:\[a = \sqrt{50}\]\[a = \sqrt{25 \times 2}\]\[a = 5\sqrt{2}\]Step 5: Since \(a = b\), \(b\) is also \(5\sqrt{2}\).Final Answer:The lengths of the legs \(a\) and \(b\) are both \(5\sqrt{2}\), which corresponds to option B.