Question

line segment on is perpendicular to line segment ml, and pn = 10.
what is the area of triangle mol?
250 square units
500 square units
300 square units
200 square units

Explanation:

Step1: Use the property of perpendicular from the center to a chord

If a line from the center of a circle (\(O\)) is perpendicular to a chord (\(ML\)), it bisects the chord. Let the length of \(ML = 2x\).

Step2: Find the length of \(MP\) using the Pythagorean theorem in \(\triangle OMP\)

In right - triangle \(OMP\), \(OM = 25\) (radius of the circle) and \(OP=ON - PN\). Since \(ON = OM = 25\) (radii of the same circle) and \(PN = 10\), then \(OP=25 - 10=15\).
By the Pythagorean theorem \(MP=\sqrt{OM^{2}-OP^{2}}\). Substitute \(OM = 25\) and \(OP = 15\) into the formula:

$$MP=\sqrt{25^{2}-15^{2}}=\sqrt{(25 + 15)(25 - 15)}=\sqrt{40\times10}=\sqrt{400}=20$$

So, \(ML=2MP = 40\) (because the perpendicular from the center to a chord bisects the chord).

Step3: Calculate the area of \(\triangle MOL\)

The area of a triangle is given by \(A=\frac{1}{2}\times base\times height\). Here, the base \(ML = 40\) and the height \(OP + PN=25\) (since \(ON\) is perpendicular to \(ML\) and \(ON\) is a radius).

$$A=\frac{1}{2}\times ML\times ON$$

Substitute \(ML = 40\) and \(ON = 25\) into the formula:

$$A=\frac{1}{2}\times40\times25= 500$$

Answer:

\(500\) square units