Question

The sides of a triangle are $10\frac{1}{2}$ , $16\frac{1}{2}$ , and 9. An angle bisector meets the side of length 9. Find the lengths 𝒙 and 𝒚.

• A. 𝒙 = 3.5 and 𝒚 = 3.5
• B. 𝒙 = 5.5 and 𝒚 = 3.5
• C. 𝒙 = 6.5 and 𝒚 = 5.5
• D. None of the above

Answer 1

The correct option is B 𝒙 = 5.5 and 𝒚 = 3.5


By the angle bisector theorem, $\frac{x}{y}=\frac{10\frac{1}{2}}{16\frac{1}{2}}$ , and 𝒚 = 𝟗 − 𝒙, so

$\frac{x}{9-x}=\frac{16.5}{10.5}$
$\therefore$ 𝟏𝟎. 𝟓𝒙 = 𝟏𝟔. 𝟓(𝟗 − 𝒙)
$\therefore$ 𝟏𝟎. 𝟓𝒙 = 𝟏𝟒𝟖. 𝟓 − 𝟏𝟔. 𝟓𝒙
$\therefore$ 𝟐𝟕𝒙 = 𝟏𝟒𝟖. 𝟓
$\therefore$ 𝒙 = 5.5

𝒚 = 𝟗 − 𝒙
$\therefore$ 𝒚 = 𝟗 − 𝟓. 𝟓
$\therefore$ 𝒚 = 3.5