Question

standard: g.gsr.4.2-dok2

parallelogram qrst has vertices q(0, 10) and s(6, 2).
which statement can be used to prove that qrst is a rhombus?

the slope of \\( \overline { q s } \\) is \\( - \frac { 4 } { 3 } \\) and the slope of \\( \overline { r t } \\) is \\( \frac { 3 } { 4 } \\).

the slope of \\( \overline { q s } \\) is \\( - \frac { 4 } { 3 } \\) and the slope of \\( \overline { r s } \\) is \\( \frac { 3 } { 4 } \\).

the length of \\( \overline { q s } \\) is 10 and the length of \\( \overline { r t } \\) is 10.

the length of \\( \overline { q s } \\) is 10 and the length of \\( \overline { r s } \\) is 10.

Explanation:

Step1: Recall the property of a rhombus

A parallelogram is a rhombus if its diagonals are perpendicular.
The formula for the slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(Q(0,10)\) and \(S(6,2)\), the slope of \(\overline{QS}\) is \(m_{QS}=\frac{2 - 10}{6-0}=\frac{-8}{6}=-\frac{4}{3}\).
If two lines with slopes \(m_1\) and \(m_2\) are perpendicular, then \(m_1\times m_2=- 1\). Let \(m_1 =-\frac{4}{3}\), if \(m_2=\frac{3}{4}\), then \(-\frac{4}{3}\times\frac{3}{4}=-1\).

Step2: Analyze each option

• Option 1:

If the slope of \(\overline{QS}\) is \(-\frac{4}{3}\) and the slope of \(\overline{RT}\) is \(\frac{3}{4}\), since \(-\frac{4}{3}\times\frac{3}{4}=-1\), the diagonals \(\overline{QS}\) and \(\overline{RT}\) are perpendicular. In a parallelogram, if the diagonals are perpendicular, then the parallelogram is a rhombus.

• Option 2:

\(\overline{QS}\) and \(\overline{RS}\) are not diagonals of the parallelogram. We need to check the relationship between diagonals (not a side and a diagonal) for the rhombus property (using the perpendicular - diagonal property for parallelograms).

• Option 3:

If the length of \(\overline{QS}\) is \(10\) and the length of \(\overline{RT}\) is \(10\), in a rectangle (a type of parallelogram), the diagonals are equal. This property is for rectangles (not for rhombuses, the property of rhombus diagonals is that they are perpendicular, not necessarily equal in length).

• Option 4:

\(\overline{QS}\) is a diagonal and \(\overline{RS}\) is a side. Their equality in length does not help in proving that the parallelogram is a rhombus. The key property for a parallelogram to be a rhombus (using diagonals) is that the diagonals are perpendicular.

Answer:

The slope of \(\overline{QS}\) is \(-\frac{4}{3}\) and the slope of \(\overline{RT}\) is \(\frac{3}{4}\).