Question

1. a right triangle is formed by the y - axis, the line ( y = 2x + 4 ), and another line. if the legs of the right triangle intersect at ( (2,8) ), what is the equation of the other line of the triangle?

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the line \(y = 2x+4\) is \(m_1 = 2\).
If two lines are perpendicular, the product of their slopes is \(- 1\), i.e., \(m_1\times m_2=-1\).
Let the slope of the other line be \(m_2\). Then \(2\times m_2=-1\), so \(m_2 =-\frac{1}{2}\).

Step2: Use the point - slope form

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(2,8)\) and \(m =-\frac{1}{2}\).
Substitute the values into the formula: \(y - 8=-\frac{1}{2}(x - 2)\).
Expand the right - hand side: \(y-8=-\frac{1}{2}x + 1\).
Add \(8\) to both sides of the equation: \(y=-\frac{1}{2}x+9\).

Answer:

\(y =-\frac{1}{2}x + 9\)