Question

question 1-10
a square pqrs has vertices p(2, 4), q(6, 8), r(10, 4) and s(6, 0). which set of steps should a student follow to prove that the diagonals of the square are perpendicular to each other?

• step 1: find the slope of diagonal pr (m₁) and slope of diagonal qs (m₂). step 2: show that m₁ + m₂ = -1.
• step 1: find the slope of diagonal pr (m₁) and slope of diagonal qs (m₂). step 2: show that m₁ = -m₂.
• step 1: find the slope of diagonal pr (m₁) and slope of diagonal qs (m₂). step 2: show that m₁ = m₂.
• step 1: find the slope of diagonal pr (m₁) and slope of diagonal qs (m₂). step 2: show that m₁ × m₂ = -1.

Explanation:

To prove two lines are perpendicular, we use the property that the product of their slopes is -1. So first, find the slopes of the two diagonals (PR and QS), then show their product is -1. Let's analyze each option:

• Option 1: Sum of slopes being -1 is not the condition for perpendicularity. Eliminate.
• Option 2: \(m_1 = -m_2\) is for lines with slopes related to being negative of each other, not perpendicular. Eliminate.
• Option 3: \(m_1 = m_2\) is for parallel lines, not perpendicular. Eliminate.
• Option 4: Step 1: Find slopes of PR (\(m_1\)) and QS (\(m_2\)). Step 2: Show \(m_1\times m_2=-1\) (which is the condition for perpendicular lines). This matches the perpendicular slope property.

Answer:

The fourth option (the one with Step 1: Find the slope of diagonal \(PR\) (\(m_1\)) and slope of diagonal \(QS\) (\(m_2\)); Step 2: Show that \(m_1\times m_2 = - 1\))