Understand and apply the Basic Proportionality Theorem (Thales Theorem) to find missing side lengths in triangles.
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Order the logical steps of the BPT proof using triangle areas.
Drag the steps into the correct logical sequence to prove the Basic Proportionality Theorem.
Historical hook about Thales.
If the corresponding angles of two triangles are equal, they are known as equiangular triangles.
A famous Greek mathematician, Thales, gave an important truth relating to two equiangular triangles:
The ratio of any two corresponding sides in two equiangular triangles is always the same.
Statement of BPT.
Theorem 6.1 (Basic Proportionality Theorem):
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Figure 6.10 for the BPT proof.
A clean geometric diagram of triangle ABC. A solid line DE is drawn parallel to the base BC, intersecting side AB at D a…
Example 1 from page 10.
If a line intersects sides and of at and respectively and is parallel to , prove that:
Exercise 6.2 Q1(i) partially solved.
In , line is drawn parallel to side . By the Basic Proportionality Theorem, the sides are divided in the same ratio, giving the equation \frac{AD}{} = \frac{}{EC}.
Substituting the given values ( cm, cm, and cm) into our proportion gives . Solving this equation by cross-multiplying yields a final missing length of cm.
Exercise 6.2 Q1(ii) stepwise solution.
Work through the calculation step-by-step to find the missing segment .
List the known values and conditions.
State the theorem equation you will use.
Plug the known values into the equation.
Show the algebraic steps to isolate AD.
State the final numerical answer with units.