13.2 Euclidean Geometry, Similarity, Circles, and Transformations

13.2 Euclidean Geometry, Similarity, Circles, and Transformations

Geometry questions reward a clean translation from a diagram to invariant relationships. Unless stated otherwise, lengths and angles are Euclidean, and a diagram should not be assumed to be drawn to scale.

---

Triangles, congruence, and similarity

The interior angles of a triangle sum to $180^\circ$ (or $\pi$ radians). Congruence is guaranteed by SSS, SAS, ASA, AAS, and the hypotenuse-leg test for right triangles. SSA is generally ambiguous, and AAA proves only similarity.

Two triangles are similar when corresponding angles agree and corresponding side lengths have a common ratio $k$. Perimeters then scale by $k$, areas by $k^2$, and volumes of similar solids by $k^3$. In a right triangle with legs $a,b$ and hypotenuse $c$, $a^2+b^2=c^2.$ The altitude to the hypotenuse creates two smaller triangles similar to the original. If the altitude divides the hypotenuse into lengths $p$ and $q$, then $c=p+q$, the altitude satisfies $h^2=pq$, and the legs satisfy $a^2=cp$ and $b^2=cq$ after consistent labeling.

For any triangle with sides $a,b,c$ opposite angles $A,B,C$, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$ is the Law of Sines, where $R$ is circumradius, and $c^2=a^2+b^2-2ab\cos C$ is the Law of Cosines. Area can be written as $K=\tfrac12 ab\sin C$ or, with semiperimeter $s=(a+b+c)/2$, Heron's formula $K=\sqrt{s(s-a)(s-b)(s-c)}.$

Four classical lines concur: medians at the centroid, perpendicular bisectors at the circumcenter, angle bisectors at the incenter, and altitudes at the orthocenter. The centroid divides every median in a $2:1$ ratio measured from the vertex.

---

Polygons, area, and solids

An $n$-gon has interior-angle sum $(n-2)180^\circ$. A regular $n$-gon has central angle $360^\circ/n$ and area $K=\tfrac12 aP$, where $a$ is its apothem and $P$ its perimeter.

Core solid formulas include prism or cylinder volume $V=Bh$, pyramid or cone volume $V=Bh/3$, sphere volume $4\pi r^3/3$, and sphere surface area $4\pi r^2$. Cavalieri's principle explains why solids with equal cross-sectional areas at every height have equal volumes.

---

Circles and power of a point

A central angle has the same degree measure as its intercepted arc. An inscribed angle has half the measure of its intercepted arc. A radius to a tangent point is perpendicular to the tangent, and tangent segments drawn from the same exterior point have equal lengths.

Power of a point consolidates chord and secant products. If two chords intersect inside a circle, $PA\cdot PB=PC\cdot PD.$ From an exterior point with two secants, external times whole is equal for both: $PA\cdot PB=PC\cdot PD.$ For a tangent $PT$ and a secant through $A,B$, $PT^2=PA\cdot PB.$ Use full secant length, not only the internal portion.

---

Vectors and transformations

For vectors $u,v$, the dot product satisfies $u\cdot v=\|u\|\|v\|\cos\theta.$ Thus $u\cdot v=0$ characterizes perpendicular nonzero vectors. In the plane, $|u_1v_2-u_2v_1|$ is the area of the parallelogram spanned by $u,v$, and half of it is the associated triangle area.

Translations, rotations, and reflections are rigid motions: they preserve lengths and angles. A rotation by $\theta$ about the origin uses $\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}.$ A dilation by factor $k$ preserves angles and multiplies lengths by $|k|$. A composition of a rigid motion and a dilation is a similarity.

Worked example

A circle has radius 10. A chord lies 6 units from the center. The perpendicular from the center bisects the chord, so half the chord, the distance 6, and radius 10 form a right triangle. Half-length is $\sqrt{10^2-6^2}=8$, hence the chord length is 16.

Common traps

Do not infer equal lengths from appearance. AAA gives similarity, not congruence. In a secant theorem, multiply the external segment by the entire secant. Surface area scales quadratically while volume scales cubically. A negative dilation factor also reverses direction through the center, but lengths scale by its absolute value.