Mathematics: Geometric and Spatial Reasoning Study Guide for the TSIA2
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Geometry and Algebra
Geometric figures can lend themselves to writing and solving algebraic equations in ways that are a little more challenging than straightforward calculations of areas or volumes. There are no new concepts in this section, just examples of perhaps a higher dose of algebra with the geometry.
Rectangular Prism
Given a right rectangular prism with the dimensions shown in the figure, what is its total surface area?
The prism has three pairs of identical sides, so we can calculate the total area by adding them.
\[\text{SA} =2(a \times b) + 2(a \times 2b) + 2(b \times2b)\] \[\text{SA} = 2ab + 4ab+ 4b^2\] \[\text{SA} = 6ab +4b^2\]Vertical Angles
Given two intersecting lines, determine the sum of the two labeled angles.
The two labeled angles are vertical angles, and are therefore congruent.
\[2x^2-4=5x-1\] \[2x^2-5x-3=0\] \[(2x+1)(x-3)=0\] \[\begin{array}{rl} 2x+1 = 0& x-3=0\\ 2x=-1 & x=3\\ x=-\frac{1}{2} & \\ \end{array}\]Rule out \(x=-\frac{1}{2}\) because it would give a negative angle, which isn’t defined in geometry. That leaves \(x=3\).
If \(x=3\), then \(5x-1 = 15-1 = 14^\circ\).
The sum of both angles is \(28^\circ\).
Sides of a Rectangle
Find the lengths of the sides of a rectangle with an area of \(24\) and a perimeter of \(22\).
If the perimeter is \(22\), we can assign the variables below to the rectangle’s sides.
Given that the area of the rectangle is \(24\), we can write this expression for area:
\[A=x(11-x) =24\] \[11x-x^2 = 24\] \[x^2-11x-24 = 0\] \[(x-3)(x-8) = 0\] \[\begin{array}{rl} x-3=0 & x-8=0\\ x=3 & x=8\\ \end{array}\]If \(x=3\), then \(11-x=8\).
If \(x=8\), then \(11-x=3\).
Either way, the sides are \(3\) and \(8\).
Angles in Polygons
A certain polygon has interior angles of \(135^\circ\). How many sides does it have? The formula for the sum of interior angles in a polygon is \((n-2) \cdot 180^\circ\) where \(n\) is the number of sides.
We are looking for \(n\), the number of sides. We can start by writing an expression for a single angle in the polygon.
The sum of interior angles is \((n-2) \cdot 180^\circ\), and there are \(n\) sides, so to find a single angle we divide the total by the number of angles. Part way through, we drop the degree symbol for simplicity.
\[\dfrac{(n-2) \cdot 180^\circ}{n} = 135^\circ\] \[(n-2) \cdot 180^\circ = 135^\circ \cdot n\] \[180n - 360 = 135n\] \[45n = 360\] \[n = 8\]All Study Guides for the TSIA2 are now available as downloadable PDFs