Question

The perimeter of $\Delta$ 𝑳𝑴𝑵 is 𝟑𝟐 𝐜𝐦. NX is the angle bisector of angle 𝑵, 𝑳𝑿 = 𝟑 𝐜𝐦, and 𝑿𝑴 = 𝟓 𝐜𝐦. Find 𝑳𝑵 and 𝑴𝑵.

• A. LN = 9 cm and MN = 15 cm
• B. LN = 15 cm and MN = 9 cm
• C. LN = 12 cm and MN = 15 cm
• D. None of the above

Answer 1

The correct option is A LN = 9 cm and MN = 15 cm


Since NX is an angle bisector of angle 𝑵, by the angle bisector theorem, 𝑿𝑳:𝑿𝑴 = 𝑳𝑵: 𝑴𝑵; thus, 𝑳𝑵: 𝑴𝑵 = 𝟑: 𝟓.

Therefore, 𝑳𝑵 = 𝟑𝒙 and 𝑴𝑵 = 𝟓𝒙 for some positive number 𝒙. The perimeter of the triangle is 𝟑𝟐 𝐜𝐦.

So, 𝑿𝑳 + 𝑿𝑴 + 𝑴𝑵 + 𝑳𝑵 = 𝟑𝟐
$\therefore$ 𝟑 + 𝟓 + 𝟑𝒙 + 𝟓𝒙 = 𝟑𝟐
$\therefore$ 𝟖 + 𝟖𝒙 = 𝟑𝟐
$\therefore$ 𝟖𝒙 = 𝟐𝟒
$\therefore$ 𝒙 = 3

$\therefore$𝟑𝒙 = 𝟑(𝟑) = 𝟗 and 𝟓𝒙 = 𝟓(𝟑) = 𝟏𝟓
$\therefore$ 𝑳𝑵 = 𝟗 𝐜𝐦 and 𝑴𝑵 = 𝟏𝟓 𝐜𝐦