Suppose the hypotenuse and its opposite vertex of an isosceles right angled triangle are \(3x+4y-4=0\) and \((2,2)\) respectively. Then, which of the following is another side of the triangle?
Show Hint
In an isosceles right triangle, the two equal sides make \(45^\circ\) angles with the hypotenuse. Use the formula
\[
\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|
\]
to find the slopes of the equal sides.
Step 1: Find the angle of the hypotenuse. The hypotenuse is \(3x+4y-4=0\), slope \(= -3/4\). In an isosceles right triangle, the two legs make \(45^\circ\) with the hypotenuse.
Step 2: Find slope of each leg. If a leg has slope m, then \(\tan 45^\circ = \left|\frac{m-(-3/4)}{1+m(-3/4)}\right| = 1\). Solving: \(m+3/4 = \pm(1-3m/4)\). Taking the minus sign: \(m+3/4 = -(1-3m/4) \Rightarrow m = -7\). Taking plus: \(m = 1/7\).
Step 3: Write equations through (2, 2). Line with slope \(-7\): \(y-2=-7(x-2) \Rightarrow 7x+y-16=0\). \[ \boxed{7x+y-16=0} \]