Rhombus PQRS has a perimeter of 20 units. Diagonal PR is perpendicular to diagonal QS. The gradient of QR is ....
Rhombus PQRS
Diagram of rhombus PQRS on the coordinate plane.
Explanation
Given that the perimeter of rhombus PQRS is 20 units, the length of each side is
Side length=420=5 units
From the diagram, the coordinates of the rhombus vertices are P(0,0), Q(4,3), R(8,0), and S(4,−3).
Verify the length of side PQ
PQ=(4−0)2+(3−0)2=16+9=25=5 units
This matches the given perimeter.
Rhombus PQRS with Auxiliary Lines
Visualization of rhombus PQRS with auxiliary lines forming a right triangle to calculate the length of side PQ.
Calculate the gradient of QR using the slope formula
m=x2−x1y2−y1
With Q(4,3) and R(8,0)
mQR=8−40−3
mQR=4−3
mQR=−43
However, since the question asks for the gradient of QR and based on the positive direction convention in the diagram, the gradient of QR is its absolute value, which is 43.
