Master the SSS and SAS similarity criteria and apply them to complex geometric and real-world problems.
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Real-world context (shadow problem) and overlapping triangles require strong visual support.
A real-life geometric diagram showing a girl standing in front of a tall lamp-post at night. The light from the lamp-pos…
Statement of Theorem 6.4.
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.
Example 5 from page 19.
Problem. Observe two triangles, and . In : , , , , and . In : , , . Find .
Statement of Theorem 6.5.
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.
Example 7 from page 20.
Problem. A girl of height is walking away from the base of a lamp-post at a speed of . If the lamp is above the ground, find the length of her shadow after .
Exercise 6.3 Q15 partial solution.
Let the height of the tower be . The triangles formed by the sun's rays are similar. We can set up a proportion using the ratio of their corresponding sides: . To solve this, we can isolate by multiplying both sides by . Calculating this gives , which means the height of the tower is meters.
Example 8 from page 21 (Medians).
Given , we know that and . Since and are medians, they divide sides and exactly in half.
Use this information to write a proof for why . Make sure to state which similarity criterion you are relying on.
Identify the theorem you are building toward.
Show how the medians mathematically relate to the proportional sides to fulfill your chosen criterion.