Question

finding the point of intersection

to find the point of intersection of the diagonals wy and xz solve the simultaneous equations.
click through the slideshow for an explanation.

Explanation:

Find the equations of the diagonals

Using the Coordinate Geometry and Gradient Formula knowledge points.

For diagonal \(WY\) connecting \(W(-3, 1)\) and \(Y(6, -2)\):

$$ m_{WY} = \frac{-2 - 1}{6 - (-3)} = \frac{-3}{9} = -\frac{1}{3} $$

Using point-slope form with \(W(-3, 1)\):

$$ y - 1 = -\frac{1}{3}(x + 3) \implies y = -\frac{1}{3}x $$

For diagonal \(XZ\) connecting \(X(3, 4)\) and \(Z(0, -5)\):

$$ m_{XZ} = \frac{-5 - 4}{0 - 3} = \frac{-9}{-3} = 3 $$

Using point-slope form with \(Z(0, -5)\):

$$ y = 3x - 5 $$

Solve the simultaneous equations

Using the Coordinate Geometry knowledge point.

Equate the two expressions for \(y\):

$$ 3x - 5 = -\frac{1}{3}x $$

Multiply the entire equation by 3 to clear the fraction:

$$ 9x - 15 = -x $$

Add \(x\) and 15 to both sides:

$$ 10x = 15 \implies x = 1.5 $$

Calculate the y-coordinate

Using the Coordinate Geometry knowledge point.

Substitute \(x = 1.5\) back into the equation for \(WY\):

$$ y = -\frac{1}{3}(1.5) = -0.5 $$

The point of intersection is \((1.5, -0.5)\).

Answer:

The point of intersection of the diagonals \(WY\) and \(XZ\) is \((1.5, -0.5)\).