Example 4 - Chapter 9 Class 9 - Propositions and their Converses (Ganita Manjari)

Transcript

Example 4 Proposition P: If two triangles have the same area, then they are congruent. Converse Q: If two triangles are congruent, then they have the same area. Determine if these statements are true or not. Justify the true statements and give a counterexample for each false statement. Let’s check if both statements are true or not Checking if Proposition P is true P: If two triangles have the same area, then they are congruent. Let’s take two triangles with the same area Consider two right triangles: Triangle 1 has perpendicular sides 8 cm and 6 cm Triangle 2 has perpendicular sides 12 cm and 4 cm Their areas are: 1/2(8)(6)=24〖" " cm〗^2 and 1/2(12)(4)=24〖" " cm〗^2. However, their side lengths differ, so they are not congruent. Thus, Proposition P is false Converse: If two triangles are congruent, then they have the same area. True. Congruent triangles have the same shape and size and therefore equal areas. Proposition false; converse true. and 1/2(12)(4)=24〖" " cm〗^2. However, their side lengths differ, so they are not congruent. Thus, Proposition P is false Checking if Converse Q is true Q: If two triangles are congruent, then they have the same area Congruent triangles have the same shape and size and therefore equal areas. Thus, Converse is true