ACT Math 7: Plane Geometry — Triangles, Polygons, and Circles

Triangle

A triangle is a polygon with three line-segment sides, three vertices, and three interior angles. In Euclidean geometry, it is a planar figure.

How is a triangle classified by side lengths?

An equilateral triangle has three equal sides, an isosceles triangle has at least two equal sides, and a scalene triangle has three different side lengths.

How is a triangle classified by angle measures?

A right triangle has one $90^\circ$ angle, an acute triangle has three angles less than $90^\circ$, and an obtuse triangle has one angle greater than $90^\circ$.

What angle relationship holds in an equilateral triangle?

All three angles are equal, so each measures $60^\circ$ because the angle sum is $180^\circ$.

Base and altitude of a triangle

A base is any chosen side of a triangle. The corresponding altitude is the perpendicular distance from the opposite vertex, called the apex, to the base or its extension.

What is the sum of the interior angles of a Euclidean triangle?

The interior angles always sum to $180^\circ$, or $\pi$ radians.

How can the third angle of a triangle be found when two angles are known?

Subtract the two known angles from $180^\circ$: $C=180^\circ-A-B$.

What is an exterior angle of a triangle, and how is it related to interior angles?

An exterior angle forms a linear pair with an interior angle, so it is supplementary to that adjacent interior angle. Its measure equals the sum of the two nonadjacent interior angles.

What trigonometric ratios are defined using a right triangle?

For an acute angle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Law of Sines

For a triangle with side lengths $a$, $b$, $c$ opposite angles $A$, $B$, $C$, respectively, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. It is useful when an angle-side opposite pair is known.

Law of Cosines

For side $c$ opposite angle $C$, $c^2=a^2+b^2-2ab\cos C$. It generalizes the Pythagorean theorem to non-right triangles.

What is a perpendicular bisector of a triangle side, and what point do the three perpendicular bisectors define?

A perpendicular bisector passes through a side's midpoint and is perpendicular to that side. The three perpendicular bisectors intersect at the circumcenter, the center of the circle passing through all three vertices.

How does the circumcenter's location indicate whether a triangle is acute, right, or obtuse?

The circumcenter lies inside an acute triangle, on the midpoint of the hypotenuse of a right triangle, and outside an obtuse triangle.

What does Thales' theorem imply about a triangle's circumcircle?

If one side of a triangle is a diameter of its circumcircle, the angle opposite that side is a right angle. Conversely, a right triangle has its hypotenuse as the diameter of its circumcircle.

Altitude and orthocenter

An altitude is a line through a vertex perpendicular to the opposite side or its extension. The three altitudes intersect at the orthocenter.

Where is the orthocenter located in an acute triangle?

In an acute triangle, all three altitudes intersect inside the triangle. In an obtuse triangle, the orthocenter lies outside.

Angle bisector and incenter

An angle bisector divides a vertex angle into two equal angles. The three angle bisectors meet at the incenter, which is the center of the incircle tangent to all three sides.

Inradius and incircle

The incircle is the largest circle contained inside a triangle and tangent to all three sides. Its radius is the inradius, commonly denoted $r$.

Median and centroid

A median connects a vertex to the midpoint of the opposite side. The three medians intersect at the centroid, which divides each median in a $2:1$ ratio measured from the vertex.

What does it mean for two triangles to be similar?

Similar triangles have equal corresponding angles and proportional corresponding side lengths. They have the same shape but may differ in size.

What is the angle-angle criterion for triangle similarity?

AA proves similarity when two corresponding angle pairs are equal.

What is the side-angle-side criterion for triangle similarity?

SAS proves similarity when two pairs of corresponding sides have the same ratio and the included angles are equal.

What is the side-side-side criterion for triangle similarity?

SSS proves similarity when all three pairs of corresponding side lengths have the same ratio.

What is the scale-factor relationship between corresponding lengths and areas of similar triangles?

If corresponding lengths have scale factor $k$, corresponding areas have scale factor $k^2$. Thus, doubling every side makes the area four times as large.

What does it mean for two triangles to be congruent?

Congruent triangles have exactly the same shape and size. Their corresponding sides and angles are equal, although the triangles may be translated, rotated, or reflected.

Which criteria can establish triangle congruence?

The standard criteria are SSS, SAS, ASA, and AAS.

Why does SSA generally not establish triangle congruence?

SSA generally does not guarantee congruence because the given information can produce more than one triangle.

What is the standard area formula for a triangle?

The area is $T=\frac{1}{2}bh$, where $b$ is the base length and $h$ is the perpendicular altitude to that base.

How can the area of a triangle be found from two sides and their included angle?

If sides $a$ and $b$ enclose angle $\gamma$, then $T=\frac{1}{2}ab\sin\gamma$.

What happens to triangle area when the included angle between two fixed sides changes?

For fixed sides $a$ and $b$, the area is $T=\frac{1}{2}ab\sin\gamma$. It is largest when $\gamma=90^\circ$ and decreases as the angle approaches $0^\circ$ or $180^\circ$.

Heron's formula

For side lengths $a$, $b$, and $c$, define the semiperimeter $s=\frac{a+b+c}{2}$. The area is $T=\sqrt{s(s-a)(s-b)(s-c)}$.

How can the area of a triangle be calculated from Cartesian coordinates?

For vertices $(x_A,y_A)$, $(x_B,y_B)$, and $(x_C,y_C)$, use $T=\frac{1}{2}|x_Ay_B+x_By_C+x_Cy_A-x_By_A-x_Cy_B-x_Ay_C|$.

What condition determines whether three positive lengths can form a nondegenerate triangle?

Each pair of side lengths must have a sum greater than the third: $a+b>c$, $a+c>b$, and $b+c>a$. Equivalently, the longest side must be shorter than the sum of the other two.

Degenerate triangle

A degenerate triangle has collinear vertices and zero area. Its side lengths satisfy equality in the triangle inequality, such as $a+b=c$, and its angles are conventionally $0^\circ$, $0^\circ$, and $180^\circ$.

Why does knowing all three side lengths make a triangle rigid?

Three side lengths determine the triangle's angles through the triangle inequalities and the law of cosines. Therefore, the triangle cannot change shape without changing a side length or breaking a joint.

Why are triangular supports used to strengthen structures?

A triangle is geometrically rigid, whereas a four-sided frame can deform into a parallelogram without changing side lengths. Adding a diagonal divides a quadrilateral into two rigid triangles.

What is triangulation of a polygon?

Triangulation partitions a polygon into nonoverlapping triangles whose edges meet along shared sides. For a simple polygon with $n$ vertices, the partition contains $n-2$ triangles and uses $n-3$ diagonals.

What is the medial or midpoint triangle of a reference triangle?

It is formed by joining the midpoints of the three sides. It is similar to the original triangle with side lengths half as large and divides the original triangle into four congruent smaller triangles.