Question

if line segment ru is considered the base of parallelogram rstu, what is the corresponding height of the parallelogram? 4.5 units 5.4 units 9.0 units 10.8 units

Explanation:

Step1: Calculate the length of RU

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For points \(R(1,1)\) and \(U(4,5)\), \(RU=\sqrt{(4 - 1)^2+(5 - 1)^2}=\sqrt{9 + 16}=\sqrt{25} = 5\).

Step2: Calculate the area of the parallelogram using another base - height pair

Take \(RS\) as the base. \(R(1,1)\) and \(S(7,0)\), \(RS=\sqrt{(7 - 1)^2+(0 - 1)^2}=\sqrt{36+1}=\sqrt{37}\approx 6.1\). The height corresponding to \(RS\) (vertical distance from \(T\) or \(U\) to \(RS\)) is not straightforward. Instead, use the formula for the area of a parallelogram \(A = base\times height\). We can also count the "unit - squares" or use the shoelace formula. Another approach: The vertical distance between the lines containing \(RU\) and \(ST\) (the height corresponding to base \(RU\)).
The slope of \(RU\) is \(m=\frac{5 - 1}{4 - 1}=\frac{4}{3}\). The equation of line \(RU\) using \(y - y_1=m(x - x_1)\) (using \(R(1,1)\)) is \(y-1=\frac{4}{3}(x - 1)\) or \(4x-3y - 1=0\).
The distance \(d\) (height) from a point (say \(S(7,0)\)) to the line \(Ax+By + C = 0\) (here \(A = 4\), \(B=-3\), \(C=-1\)) is given by \(d=\frac{\vert Ax_0+By_0 + C\vert}{\sqrt{A^2 + B^2}}\).
Substitute \(x_0 = 7\), \(y_0 = 0\): \(d=\frac{\vert4\times7-3\times0 - 1\vert}{\sqrt{4^2+( - 3)^2}}=\frac{\vert28 - 1\vert}{5}=\frac{27}{5}=5.4\).

Answer:

5.4 units