Question

point n is between points m and r, mr=25, mn=x²-3x, and nr = 15. find mn.

Explanation:

Step1: Use segment addition postulate

Since \( N \) is between \( M \) and \( R \), we have \( MR = MN + NR \). Substituting the given values: \( 25=(x^{2}-3x)+15 \)

Step2: Simplify the equation

Subtract 25 from both sides: \( x^{2}-3x + 15-25 = 0 \), which simplifies to \( x^{2}-3x - 10 = 0 \)

Step3: Factor the quadratic equation

Factor \( x^{2}-3x - 10 \): \( (x - 5)(x + 2)=0 \)

Step4: Solve for \( x \)

Set each factor to zero: \( x - 5 = 0 \) gives \( x = 5 \); \( x + 2 = 0 \) gives \( x=-2 \)

Step5: Check valid \( x \)

Since length can't be negative, we check \( x = 5 \) ( \( x=-2 \) would give negative \( MN \)). Substitute \( x = 5 \) into \( MN=x^{2}-3x \): \( MN = 5^{2}-3\times5=25 - 15 = 10 \)

Answer:

\( 10 \)