Questions: Using only the given lines, explain the stated problem, how many triangles are there in this figure? (hint: List all types of the form ABC, ADE, and so on and sum)

Using only the given lines, explain the stated problem, how many triangles are there in this figure? (hint: List all types of the form ABC, ADE, and so on and sum)

Transcript text: Using only the given lines, explain the stated problem, how many triangles are there in this figure? (hint: List all types of the form ABC, ADE, and so on and sum)

Solution

Solution Steps

Step 1: List all possible sets of three vertices

The possible sets of three vertices are: ABC, ABD, ABE, ABF, ACD, ACE, ACF, ADE, ADF, AEF, BCD, BCE, BCF, BDE, BDF, BEF, CDE, CDF, CEF, DEF

Step 2: Identify valid triangles

The valid triangles are those where the three vertices are connected by lines. ABC, ACD, ADE, AEF, ABF, ABE BCD, BCE, BCF, BDE, BDF, BEF CDE, CDF, CEF, DEF

The invalid sets of vertices are: ACF, ADF.

Valid triangles are: ABC, ACD, ADE, AEF BCD, BCE, BDE, BEF CDF, CEF, DEF

However, due to intersecting lines, additional triangles are formed. FAB, FBC, FCD, FDA EAB, EBC, ECD, EDA

Triangles formed by F: FAB, FBC, FCD, FDA

Triangles formed by E: EAB, EBC, ECD, EDA

Step 3: Count the valid triangles

Listing the triangles: ABC, ACD, ADE, AEF FAB, FBC, FCD, FDA EAB, EBC, ECD, EDA

There are 12 triangles in total.