Question

1. le segment de droite pt

le point p est le point dintersection des segments de droite wn et st représentés ci-dessous dans le plan cartésien.

• léquation associée au segment de droite st est \\(y = -\frac{1}{2}x + 200\\).
• le point t est lun des points de laxe des x.

au centième dunité près, quelle est la mesure du segment de droite pt?

Explanation:

Find coordinates of point T

Point \(T\) lies on the \(x\)-axis, so its \(y\)-coordinate is \(0\).
Using the equation of the line \(ST\):

$$y = -\frac{1}{2}x + 200$$

Set \(y = 0\) to find the \(x\)-coordinate of \(T\):

$$0 = -\frac{1}{2}x + 200 \implies \frac{1}{2}x = 200 \implies x = 400$$

Thus, the coordinates of \(T\) are \((400, 0)\).

Determine equation of line WN

The line segment \(WN\) is perpendicular to \(ST\).
The slope of \(ST\) is \(m_1 = -\frac{1}{2}\).
The slope \(m_2\) of a perpendicular line satisfies:

$$m_1 \cdot m_2 = -1 \implies -\frac{1}{2} \cdot m_2 = -1 \implies m_2 = 2$$

Using the point \(W(0, 45)\), which is the \(y\)-intercept, the equation of line \(WN\) is:

$$y = 2x + 45$$

Find intersection point P

Point \(P\) is the intersection of lines \(ST\) and \(WN\).
Set their equations equal to each other:

$$2x + 45 = -\frac{1}{2}x + 200$$

Multiply the entire equation by \(2\) to clear the fraction:

$$4x + 90 = -x + 400$$

$$5x = 310 \implies x = 62$$

Substitute \(x = 62\) back into the equation for \(WN\) to find \(y\):

$$y = 2(62) + 45 = 124 + 45 = 169$$

Thus, the coordinates of \(P\) are \((62, 169)\).

Calculate length of segment PT

Use the distance formula between \(P(62, 169)\) and \(T(400, 0)\):

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

$$d = \sqrt{(400 - 62)^2 + (0 - 169)^2}$$

$$d = \sqrt{(338)^2 + (-169)^2}$$

$$d = \sqrt{114244 + 28561} = \sqrt{142805}$$

$$d \approx 377.8955$$

Round to the nearest hundredth

Rounding \(377.8955\) to two decimal places gives:

$$d \approx 377.90$$

Answer:

\(377.90\)