Question

with the information given below, determine how triangle tud can be shown to be similar to triangle suy.
tu = 6 cm, du = 9 cm, st = 4 cm, yd = 6 cm
note: picture not drawn to scale.
a. the triangles are similar by sss
b. the triangles are similar by aa
c. the triangles are not similar to each other.
d. the triangles are similar by sas

Explanation:

Step1: Calculate \(SU\) and \(YU\)

We know that \(SU = TU+ST\). Given \(TU = 6\mathrm{cm}\) and \(ST = 4\mathrm{cm}\), so \(SU=6 + 4=10\mathrm{cm}\).
Also, \(YU=DU + YD\). Given \(DU = 9\mathrm{cm}\) and \(YD = 6\mathrm{cm}\), so \(YU=9 + 6=15\mathrm{cm}\).

Step2: Check the ratio of sides

We check the ratio of \(\frac{TU}{SU}\) and \(\frac{DU}{YU}\).
\(\frac{TU}{SU}=\frac{6}{10}=\frac{3}{5}\), \(\frac{DU}{YU}=\frac{9}{15}=\frac{3}{5}\).
And \(\angle TUD=\angle SUY\) (common angle).

Step3: Apply the SAS similarity criterion

Since \(\frac{TU}{SU}=\frac{DU}{YU}\) and \(\angle TUD=\angle SUY\), by the Side - Angle - Side (SAS) similarity criterion, \(\triangle TUD\sim\triangle SUY\).

Answer:

D. The triangles are similar by SAS.