Given that lines l 𝑙 and n 𝑛 are perpendicular, which of the following is part of the proof that l 𝑙 and n 𝑛 have slopes that are opposite reciprocals? (1 point) Responses △ABC∼△EDA △ 𝐴 𝐵 𝐶 ∼ △ 𝐸 𝐷 𝐴 triangle cap A cap b cap c approximates triangle cap e cap d cap A ∠ABC≅∠DAE ∠ 𝐴 𝐵 𝐶 ≅ ∠ 𝐷 𝐴 𝐸 ∠ABC≅∠DAE ∠ 𝐴 𝐵 𝐶 ≅ ∠ 𝐷 𝐴 𝐸 The slope of line l 𝑙 is BCCA 𝐵 𝐶 𝐶 𝐴 and the slope of line n 𝑛 is DEAD 𝐷 𝐸 𝐴 𝐷 . The slope of line l is cap b cap c over cap c cap A and the slope of line n is cap d cap e over cap A cap d. BCCA=−DEAD

Question

GPT-4o mini · Answer

In a proof that lines $ l $ and $ n $ are perpendicular and have slopes that are opposite reciprocals, it’s essential to demonstrate the relationship between their slopes. The relevant parts of your response that contribute to this proof would likely include the following components: 1. The statement indicating that the slopes of the lines are given as $ \frac{BC}{CA} $ for line $ l $ and $ \frac{DE}{AD} $ for line $ n $. This helps establish what the slopes are. 2. The conclusion that follows, stating that $ \frac{BC}{CA} = -\frac{DE}{AD} $. This shows that the product of the slopes is $-1$, which is a defining characteristic of perpendicular lines. So, the response that essentially highlights these elements: "The slope of line $ l $ is $ \frac{BC}{CA} $ and the slope of line $ n $ is $ \frac{DE}{AD} $. The slopes are related by $ BCC_A = -DEAD $ (though you may need to correct the notation to denote it properly)." Thus, the core part of your proof would be the relationship showing that the slopes of the two lines are negative reciprocals of each other, which confirms that the lines are perpendicular.