Let triangle ABC be a triangle such that angle ACB is 135 degrees.
Prove that AB^2 = AC^2 + BC^2 - (Root 2) x AC x BC
To prove that AB^2 = AC^2 + BC^2 - √2 * AC * BC, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, we have the following relationship:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, we are given that angle ACB is 135 degrees, so we can label the sides and angle as follows:
C
/ \
/ \
a / \ b
/ \
A---------B
Using the Law of Cosines, we have:
AB^2 = AC^2 + BC^2 - 2 * AC * BC * cos(ACB)
Since cos(135) = -√2/2, we can substitute this value into the equation:
AB^2 = AC^2 + BC^2 - 2 * AC * BC * (-√2/2)
= AC^2 + BC^2 + √2 * AC * BC
Hence, we have proven that AB^2 = AC^2 + BC^2 - √2 * AC * BC in the given triangle.