Use the two established similarities to prove the product identity.

01 / Standard#Your turn

DG\cdot DF=DB\cdot EF

Official paper · jm01-2025 · II.4(c) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express DE² using each similarity.

Hint 2

Equate the two products.

Worked solution

The first similarity gives a mean-proportional relation.
$\frac{DE}{DF}=\frac{DG}{DE}\implies DE^2=DG\cdot DF$
The second similarity expresses the same square.
$\frac{DE}{DB}=\frac{EF}{DE}\implies DE^2=DB\cdot EF$
Both expressions equal DE².
$DG\cdot DF=DB\cdot EF$

DG·DF=DB·EF.

Checks and common pitfalls: Ratios must follow the vertex correspondence, not visual side lengths.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The first similarity gives a mean-proportional relation.
$\frac{DE}{DF}=\frac{DG}{DE}\implies DE^2=DG\cdot DF$
The second similarity expresses the same square.
$\frac{DE}{DB}=\frac{EF}{DE}\implies DE^2=DB\cdot EF$
Both expressions equal DE².
$DG\cdot DF=DB\cdot EF$

Think first. Reveal a hint when the class is ready.

2025 JM01