What is the gradient of a line perpendicular to y = -2x + 5?

The gradient of a line perpendicular to \( y = -2x + 5 \) is \(\frac{1}{2}\).

To understand why, let's start by looking at the gradient of the given line. The equation \( y = -2x + 5 \) is in the form \( y = mx + c \), where \( m \) represents the gradient. Here, the gradient \( m \) is \(-2\).

When two lines are perpendicular, the product of their gradients is \(-1\). This is a key property of perpendicular lines. So, if the gradient of one line is \( m \), the gradient of the line perpendicular to it will be \(-\frac{1}{m}\).

In this case, the gradient of the given line is \(-2\). To find the gradient of the perpendicular line, we take the negative reciprocal of \(-2\). The reciprocal of \(-2\) is \(-\frac{1}{2}\), and the negative of that is \(\frac{1}{2}\). Therefore, the gradient of the line perpendicular to \( y = -2x + 5 \) is \(\frac{1}{2}\).

This means that if you were to draw a line with a gradient of \(\frac{1}{2}\), it would intersect the original line at a right angle. This concept is very useful in geometry and various applications, such as finding the shortest distance between a point and a line or constructing perpendicular bisectors in coordinate geometry.