12.3 The Pythagorean Theorem, Geometric Similarity, & Indirect Measurement

12.3 The Pythagorean Theorem, Geometric Similarity, & Indirect Measurement

Right-triangle mathematics is the backbone of the construction, carpentry, manufacturing, and surveying trades, and it carries real weight on TABE. The published DRC content specification puts three right-triangle skills on the test:

• Level D — 8.G.7: apply the Pythagorean theorem to find unknown side lengths in right triangles, in two and three dimensions.
• Level D — 8.G.8: apply the Pythagorean theorem to find the distance between two points in a coordinate system.
• Level A — G.SRT.5: use congruence and similarity criteria for triangles to solve problems and prove relationships in figures.

Note what is not there: sine, cosine, and tangent. Right-triangle trigonometry does not appear in the DRC content specification for any TABE level, and the official Level A reference sheet provides no trigonometric ratios. This section therefore teaches the Pythagorean theorem, its converse, and similarity as core tested content, and treats SOH CAH TOA as a clearly labeled bonus appendix for learners who are heading on to the GED or HiSET, where it is tested.

Reference-sheet reality check. The Pythagorean theorem $a^2 + b^2 = c^2$ is printed on the Level D and Level A reference sheets. The distance formula is not — which is exactly why 8.G.8 asks you to get distance from the Pythagorean theorem instead.

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The Pythagorean Theorem

In any right triangle (a triangle containing one $90^\circ$ angle), the square of the longest side (the hypotenuse, $c$) equals the sum of the squares of the two shorter sides (the legs, $a$ and $b$):

$\mathbf{a^2 + b^2 = c^2}$

Right Triangle Anatomy:

                 ▲ B
                 |\
                 | \
                 |  \   Hypotenuse (c)
          Leg a  |   \  (Longest side, strictly opposite 90° angle)
                 |    \
                 |     \
               C |_     \ A
                 |_|_____▼
                   Leg b

Solving for Unknown Sides

• Solving for the Hypotenuse ($c$): Take the square root of the sum of the squared legs: $\mathbf{c = \sqrt{a^2 + b^2}}$
• Solving for a Missing Leg ($a$ or $b$): Subtract the square of the known leg from the square of the hypotenuse before taking the square root: $\mathbf{a = \sqrt{c^2 - b^2}} \qquad | \qquad \mathbf{b = \sqrt{c^2 - a^2}}$

[!CAUTION] Addition vs. Subtraction Trap: When solving for the hypotenuse, you add the squares ($a^2 + b^2$). When solving for a missing leg, you must subtract ($c^2 - b^2$). Adding the squares when finding a leg produces an impossible leg length longer than the hypotenuse.

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Common Pythagorean Triples & Scaled Multiples

A Pythagorean triple is a set of three positive integers $(a, b, c)$ that exactly satisfy $a^2 + b^2 = c^2$. Recognizing these primitive triples and their scaled multiples eliminates tedious manual calculations on the TABE:

TABE Speed Tip: If a right triangle has leg $a = 15$ and hypotenuse $c = 25$, divide both by $5$ to reveal the base ratio $(?, 3, 5) \implies 4$. Multiplying back by $5$ immediately gives missing leg $b = 4 \times 5 = \mathbf{20}$.

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The Converse of the Pythagorean Theorem

The Converse of the Pythagorean Theorem determines whether three given side lengths form a right, acute, or obtuse triangle. First, arrange the three sides in ascending order so that $a \le b < c$ (where $c$ is the longest side). Verify the Triangle Inequality Theorem ($a + b > c$), then compare $a^2 + b^2$ with $c^2$:

Triangle Classification Rules (c = Longest Side):

1. Right Triangle:       a² + b² = c²  (Contains exactly one 90° angle)
2. Acute Triangle:       a² + b² > c²  (All three angles are < 90°)
3. Obtuse Triangle:      a² + b² < c²  (Contains one angle > 90°)

Classification Examples

• Side lengths $7, 10, 12$: $a^2 + b^2 = 7^2 + 10^2 = 49 + 100 = 149 \qquad | \qquad c^2 = 12^2 = 144$ Since $149 > 144$ ($a^2 + b^2 > c^2$), the triangle is Acute.
• Side lengths $6, 8, 11$: $a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100 \qquad | \qquad c^2 = 11^2 = 121$ Since $100 < 121$ ($a^2 + b^2 < c^2$), the triangle is Obtuse.

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Geometric Similarity & Indirect Measurement

Two geometric figures are similar ($\sim$) if they share the exact same shape but differ in size:

1. Corresponding angles are congruent ($\cong$): $\angle A = \angle D, \angle B = \angle E, \angle C = \angle F$.
2. Corresponding side lengths are proportional: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$

Applied Shadow Reckoning Problem

Problem: A $6\text{-foot}$ tall construction worker stands next to a vertical utility pole on level ground. The worker casts an $8\text{-foot}$ shadow, while at the exact same moment the utility pole casts a $36\text{-foot}$ shadow. How tall is the utility pole?

$\frac{\text{Worker Height}}{\text{Worker Shadow}} = \frac{\text{Pole Height}}{\text{Pole Shadow}} \implies \frac{6}{8} = \frac{h}{36}$ $8h = 6 \times 36 \implies 8h = 216 \implies h = \frac{216}{8} = \mathbf{27\text{ feet}}$

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Standard 8.G.8: Distance Between Two Points Without the Distance Formula

TABE does not give you the distance formula, and it does not need to. Any two points on a coordinate grid are the endpoints of the hypotenuse of a right triangle whose legs are the horizontal and vertical gaps between them.

Procedure:

1. Find the horizontal leg: $|x_2 - x_1|$.
2. Find the vertical leg: $|y_2 - y_1|$.
3. Apply $a^2 + b^2 = c^2$ and take the positive square root.

Worked example. Find the distance between $A(-4, 2)$ and $B(8, -3)$.

$\text{horizontal leg} = |8 - (-4)| = 12 \qquad \text{vertical leg} = |-3 - 2| = 5$ $c = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = \mathbf{13}$

Because $(5, 12, 13)$ is a Pythagorean triple, this one can be read off without any arithmetic once the legs are known — one more reason the triples table above is worth memorizing.

Three-dimensional extension (also 8.G.7). To find the longest straight rod that fits diagonally inside a rectangular box, apply the theorem twice. For a crate $12 \times 9 \times 8$ inches: the base diagonal is $\sqrt{12^2 + 9^2} = \sqrt{225} = 15$ inches, and the space diagonal is $\sqrt{15^2 + 8^2} = \sqrt{289} = \mathbf{17}$ inches.

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Appendix (Beyond TABE): Right Triangle Trigonometry (SOH CAH TOA)

This subsection is not TABE content. No DRC content specification for TABE Levels E, M, D, or A includes a trigonometric-ratio standard, and the Level A reference sheet supplies no trig values. Skip it if TABE is your only goal. Work it if your next step is the GED Mathematical Reasoning or HiSET Mathematics test, both of which do assess right-triangle trigonometry.

Trigonometry evaluates the mathematical relationship between the acute angles and side ratios of a right triangle. Relative to a chosen reference angle ($\theta$):

• Hypotenuse ($\text{Hyp}$): The longest side, directly opposite the $90^\circ$ right angle.
• Opposite ($\text{Opp}$): The leg across from reference angle $\theta$.
• Adjacent ($\text{Adj}$): The leg touching reference angle $\theta$ (excluding the hypotenuse).

Trigonometric Reference Framework:

                 ▲ 
                 |\
                 | \
  Opposite Side  |  \   Hypotenuse (Hyp)
      (Opp)      |   \ 
                 |    \
               |_|_____\ θ
                 Adjacent (Adj)

The Three Fundamental Ratios: SOH CAH TOA

$\mathbf{\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}} \qquad | \qquad \mathbf{\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}} \qquad | \qquad \mathbf{\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}}$

[!TIP] Memory Mnemonic: "Some Old Horse Caught Another Horse Taking Oats Away" ($\mathbf{S}OH - \mathbf{C}AH - \mathbf{T}OA$).

Angle of Elevation vs. Angle of Depression

• Angle of Elevation: The angle formed looking upward from a horizontal line of sight to an elevated object.
• Angle of Depression: The angle formed looking downward from a horizontal line of sight to an object below.
• Geometric Rule: Because horizontal sightlines are parallel, the angle of elevation from ground point $A$ to top point $B$ equals the angle of depression from point $B$ to point $A$ (alternate interior angles).

$\text{Angle of Elevation} = \text{Angle of Depression}$

Worked Trigonometry Example: Surveying Tower Height

Problem: A surveyor stands $120\text{ ft}$ from the base of a communications tower on level ground. The angle of elevation to the top of the tower is $32^\circ$. Using $\tan(32^\circ) \approx 0.6249$, what is the height of the tower ($h$)? $\tan(32^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{120}$ $h = 120 \times \tan(32^\circ) = 120 \times 0.6249 = \mathbf{74.988\text{ ft}} \approx \mathbf{75.0\text{ ft}}$