Question

if line segment ab measures approximately 8.6 units and is considered the base of parallelogram abcd, what is the approximate corresponding height of the parallelogram? round to the nearest tenth. 3.7 units 4.1 units 4.8 units 5.6 units

Explanation:

Step1: Find the area of the parallelogram using the formula for the area of a triangle

The area of parallelogram \(ABCD\) is twice the area of \(\triangle ABD\) (or \(\triangle ABC\), \(\triangle BCD\), \(\triangle ACD\)).
We can use the formula for the area of a triangle \(A=\frac{1}{2}\times base\times height\).
Counting the units on the coordinate - grid:
The base of \(\triangle ABD\) (if we consider the horizontal or vertical distance for simplicity in area calculation) can be found by using the formula for the area of a triangle.
Another way is to use the shoelace formula. But a simpler approach is to enclose the parallelogram in a rectangle.
The rectangle that encloses the parallelogram has vertices \((1,1)\), \((1,8)\), \((13,8)\), \((13,1)\). The area of the rectangle \(A_{r}=(13 - 1)\times(8 - 1)=84\) square units.
The non - parallelogram parts (the four right - triangles around the parallelogram):
Two of the right - triangles have legs of lengths \(2\) and \(4\) (area of each \(\frac{1}{2}\times2\times4 = 4\) square units) and two have legs of lengths \(3\) and \(6\) (area of each \(\frac{1}{2}\times3\times6=9\) square units).
The area of the parallelogram \(A = 84-(2\times4 + 2\times9)=84-(8 + 18)=35\) square units.

Step2: Use the formula for the area of a parallelogram \(A = b\times h\)

We know that the base \(b = AB\approx8.6\) units and the area \(A\approx35\) square units.
From the formula \(A=b\times h\), we can solve for \(h\): \(h=\frac{A}{b}\).
Substitute \(A = 35\) and \(b = 8.6\) into the formula: \(h=\frac{35}{8.6}\approx4.1\) units.

Answer:

\(4.1\) units