Worked Examples: Geometry, Trig, Matrices & Complex Numbers

Example 1: Law of Cosines (Hard)

In triangle $PQR$, $PQ = 8$, $QR = 11$, and $\angle Q = 60°$. What is the length of $PR$?

Solution: Apply Law of Cosines with the included angle at $Q$: $PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)\cos Q$ $PR^2 = 64 + 121 - 2(8)(11)\cos 60°$ $PR^2 = 185 - 176(0.5) = 185 - 88 = 97$ $PR = \sqrt{97} \approx 9.85$

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Example 2: Complex Number Multiplication (Medium)

What is $(5 - 3i)(2 + i)$?

Solution: FOIL and replace $i^2$ with $-1$: $= 10 + 5i - 6i - 3i^2 = 10 - i - 3(-1) = 10 - i + 3 = 13 - i$

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Example 3: Matrix Determinant (Medium)

What is the determinant of $\begin{pmatrix} 7 & 3 \\ 2 & 5 \end{pmatrix}$?

Solution: $\det = ad - bc = (7)(5) - (3)(2) = 35 - 6 = 29$.

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Example 4: Coordinate Geometry — Perpendicular Lines (Medium)

Line $\ell$ passes through $(1, 4)$ and $(3, 10)$. What is the slope of any line perpendicular to $\ell$?

Solution: Slope of $\ell$: $m = \frac{10 - 4}{3 - 1} = \frac{6}{2} = 3$.

Perpendicular slope is the negative reciprocal: $m_{\perp} = -\frac{1}{3}$.

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Example 5: Powers of $i$ (Easy-Medium)

What is $i^{42}$?

Solution: Divide 42 by 4: $42 = 4(10) + 2$. Remainder is 2, so $i^{42} = i^2 = -1$.

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Example 6: Similar Triangles with Area (Hard)

Triangle $ABC$ is similar to triangle $DEF$. The sides of $ABC$ measure 6, 8, and 10. If the longest side of $DEF$ measures 15, what is the area of $DEF$?

Solution: The side ratio is $\frac{15}{10} = \frac{3}{2}$.

Area of $ABC$: Since 6-8-10 is a right triangle (Pythagorean triple), area = $\frac{1}{2}(6)(8) = 24$.

The area ratio is the square of the side ratio: $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$.

Area of $DEF = 24 \cdot \frac{9}{4} = 54$.

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Example 7: 2×2 Matrix Multiplication (Medium-Hard)

If $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & -1 \\ 2 & 5 \end{pmatrix}$, what is the entry in row 1, column 2 of $AB$?

Solution: Row 1 of $A$ is $(2, 1)$, Column 2 of $B$ is $(-1, 5)$.

Dot product: $2(-1) + 1(5) = -2 + 5 = 3$.