Similarity Criteria: SSS And SAS

Beyond Angles: Using Sides for Similarity

What if you only know the side lengths of two triangles? Do you need to check all the angles to see if they're similar? The answer is no, thanks to a specific similarity rule.

Problem. Observe two triangles, $\Delta ABC$ and $\Delta PQR$.
In $\Delta ABC$: $AB = 3.8$, $BC = 6$, $CA = 3\sqrt{3}$, $\angle A = 80^\circ$, and $\angle B = 60^\circ$.
In $\Delta PQR$: $PQ = 12$, $QR = 7.6$, $PR = 6\sqrt{3}$.
Find $\angle P$.

Blending Sides and Angles

We know we can prove triangles similar by checking all three angles (AAA) or all three sides (SSS). But what if we have a mix of information? This brings us to the SAS criterion.

Problem. A girl of height $90 \text{ cm}$ is walking away from the base of a lamp-post at a speed of $1.2 \text{ m/s}$. If the lamp is $3.6 \text{ m}$ above the ground, find the length of her shadow after $4 \text{ seconds}$.