Question

talia is writing a coordinate proof to show that the diagonals of a square are perpendicular. she begins by assigning coordinates to the vertices of a square as shown. which sentence describes what talia should do to show that the diagonals of the square are perpendicular? show that the slope of ( overline{sq} ) is 1 and the slope of ( overline{pr} ) is -1. show that ( sq = pr = asqrt{2} ). show that ( overline{sq} ) and ( overline{pr} ) intersect at ( (\frac{a}{2}, \frac{a}{2}) ). show that the midpoint of ( overline{sq} ) is the same as the midpoint of ( overline{pr} ).

Explanation:

Step1: Recall the slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For two points \((x_1,y_1)\) and \((x_2,y_2)\) on a line.

Step2: Calculate the slope of \(SQ\)

For points \(S(0,0)\) and \(Q(a,a)\), using the slope formula \(m_{SQ}=\frac{a - 0}{a - 0}=1\).

Step3: Calculate the slope of \(PR\)

For points \(P(0,a)\) and \(R(a,0)\), using the slope formula \(m_{PR}=\frac{0 - a}{a - 0}=- 1\).

Step4: Recall the condition for perpendicular lines

Two lines with slopes \(m_1\) and \(m_2\) are perpendicular if \(m_1\times m_2=-1\). Here \(m_{SQ}\times m_{PR}=1\times(-1)=-1\)

Answer:

Show that the slope of \(\overline{SQ}\) is \(1\) and the slope of \(\overline{PR}\) is \(-1\).