Question

mn is a diameter of ⊙p
lm = 2x
ln = 3x
*not drawn to scale
what is mn expressed in terms of x?
a. 4x
b. 5x
c. x√13
d. 2x√13

Explanation:

Step1: Apply the Pythagorean theorem

Since $MN$ is a diameter of the circle $\odot P$, $\angle MLN = 90^{\circ}$ (angle - in - a - semi - circle property). In right - triangle $LMN$, by the Pythagorean theorem $MN^{2}=LM^{2}+LN^{2}$. Given $LM = 2x$ and $LN = 3x$.
So, $MN^{2}=(2x)^{2}+(3x)^{2}$.

Step2: Expand and simplify the right - hand side

$(2x)^{2}+(3x)^{2}=4x^{2}+9x^{2}=13x^{2}$.

Step3: Solve for $MN$

Taking the square root of both sides, $MN=\sqrt{13x^{2}}$. Since $MN>0$, $MN = x\sqrt{13}$.

Answer:

C. $x\sqrt{13}$