SSC CAPFS and DP - triangle geometry - centres of triangle, congruence and similarity of triangles E-Book Study Material

Triangle Geometry

Centres of a Triangle

A triangle, a fundamental geometric shape, possesses several significant points known as its 'centres'. These centres are formed by the intersection of specific lines within the triangle, such as medians, altitudes, angle bisectors, and perpendicular bisectors. Understanding these centres is crucial for solving various geometry problems. Let's explore the most important ones: the centroid, orthocentre, incenter, and circumcenter.

1. Centroid (G)

The centroid is the point of intersection of the medians of a triangle. A median is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has three medians, and they all intersect at a single point, the centroid. The centroid is also the center of mass or center of gravity of a uniform triangular lamina.

Properties of the Centroid:

The centroid divides each median in a 2:1 ratio. The segment from the vertex to the centroid is twice as long as the segment from the centroid to the midpoint of the opposite side. If AD is a median and G is the centroid, then AG : GD = 2 : 1.
The three medians divide the triangle into six smaller triangles of equal area.

Calculation: If the coordinates of the vertices of a triangle are $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$, then the coordinates of the centroid G are given by: $$G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$$

2. Orthocentre (H)

The orthocentre is the point of intersection of the altitudes of a triangle. An altitude is a line segment drawn from a vertex perpendicular to the opposite side. Every triangle has three altitudes, which are concurrent at the orthocentre.

Properties of the Orthocentre:

The location of the orthocentre depends on the type of triangle:
- In an acute-angled triangle, the orthocentre lies inside the triangle.
- In a right-angled triangle, the orthocentre coincides with the vertex of the right angle.
- In an obtuse-angled triangle, the orthocentre lies outside the triangle.
In an equilateral triangle, the orthocentre, centroid, incenter, and circumcenter all coincide at the same point.

Calculation: Calculating the orthocentre's coordinates directly from vertex coordinates is more complex and usually involves finding the equations of two altitudes and their intersection point.

3. Incenter (I)

The incenter is the point of intersection of the angle bisectors of a triangle. An angle bisector is a line segment that divides an angle into two equal parts. The incenter is equidistant from the three sides of the triangle. This distance is the radius of the inscribed circle (incircle), and the incenter is the center of this circle.

Properties of the Incenter:

The incenter is the center of the incircle, which is the largest circle that can be inscribed within the triangle, touching all three sides.
The incenter is always located inside the triangle.
The distance from the incenter to each side is the inradius (r).

Calculation: If the coordinates of the vertices are $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$ and the lengths of the opposite sides are $a$, $b$, $c$ respectively, then the coordinates of the incenter I are: $$I = \left(\frac{ax_1 + bx_2 + cx_3}{a + b + c}, \frac{ay_1 + by_2 + cy_3}{a + b + c}\right)$$

The inradius $r$ can be calculated using the formula: $Area = r \times s$, where $s$ is the semi-perimeter of the triangle ($s = \frac{a+b+c}{2}$).

4. Circumcenter (O)

The circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle. A perpendicular bisector of a side is a line that is perpendicular to the side and passes through its midpoint. The circumcenter is equidistant from the three vertices of the triangle. This distance is the radius of the circumscribed circle (circumcircle), and the circumcenter is the center of this circle.

Properties of the Circumcenter:

The circumcenter is the center of the circumcircle, which passes through all three vertices of the triangle.
The location of the circumcenter depends on the type of triangle:
- In an acute-angled triangle, the circumcenter lies inside the triangle.
- In a right-angled triangle, the circumcenter is the midpoint of the hypotenuse.
- In an obtuse-angled triangle, the circumcenter lies outside the triangle.
The distance from the circumcenter to each vertex is the circumradius (R).

Calculation: Similar to the orthocentre, calculating the circumcenter's coordinates involves finding the intersection of two perpendicular bisectors.

The circumradius $R$ can be calculated using the formula: $R = \frac{abc}{4 \times Area}$.

Memory Tip for Centres:

Centroid: Intersection of Medians (C-M)
Orthocentre: Intersection of Altitudes (O-A)
Incenter: Intersection of Angle Bisectors (I-B)
Circumcenter: Intersection of Perpendicular Bisectors (C-B)

Also remember that for an equilateral triangle, all these centres coincide. For right-angled triangles, the orthocentre is the right-angle vertex, and the circumcenter is the midpoint of the hypotenuse.

Congruence of Triangles

Two triangles are said to be congruent if they are identical in shape and size. This means that all corresponding sides and all corresponding angles are equal. If triangle ABC is congruent to triangle PQR, we denote it as $\triangle ABC \cong \triangle PQR$. This implies that:

Side AB = Side PQ
Side BC = Side QR
Side CA = Side RP
Angle A = Angle P
Angle B = Angle Q
Angle C = Angle R

To prove that two triangles are congruent, we don't need to show that all six corresponding parts are equal. There are specific congruence criteria (or postulates/theorems) that allow us to establish congruence with fewer conditions.

Congruence Criteria:

1. SSS (Side-Side-Side) Congruence Criterion

If three sides of one triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent.
If $AB = PQ$, $BC = QR$, and $CA = RP$, then $\triangle ABC \cong \triangle PQR$.

2. SAS (Side-Angle-Side) Congruence Criterion

If two sides and the included angle of one triangle are equal to the two corresponding sides and the included angle of another triangle, then the two triangles are congruent. The angle must be between the two sides.
If $AB = PQ$, $\angle B = \angle Q$, and $BC = QR$, then $\triangle ABC \cong \triangle PQR$.

3. ASA (Angle-Side-Angle) Congruence Criterion

If two angles and the included side of one triangle are equal to the two corresponding angles and the included side of another triangle, then the two triangles are congruent. The side must be between the two angles.
If $\angle B = \angle Q$, $BC = QR$, and $\angle C = \angle R$, then $\triangle ABC \cong \triangle PQR$.

4. AAS (Angle-Angle-Side) Congruence Criterion

If two angles and a non-included side of one triangle are equal to the two corresponding angles and the corresponding non-included side of another triangle, then the two triangles are congruent. This criterion is derived from ASA because if two angles are equal, the third angle must also be equal (since the sum of angles in a triangle is 180 degrees).
If $\angle B = \angle Q$, $\angle C = \angle R$, and $AC = PR$, then $\triangle ABC \cong \triangle PQR$. (Note: AC and PR are non-included sides here).

5. RHS (Right Angle-Hypotenuse-Side) Congruence Criterion

If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, then the two triangles are congruent. This applies only to right-angled triangles.
If $\triangle ABC$ and $\triangle PQR$ are right-angled triangles with $\angle B = \angle Q = 90^\circ$, and $AC = PR$ (hypotenuses) and $BC = QR$ (a side), then $\triangle ABC \cong \triangle PQR$.

Key takeaway for Congruence: To prove congruence, you need to establish equality for three corresponding parts, with at least one of them being a side. The SSS, SAS, ASA, AAS, and RHS criteria are your tools. Always ensure the corresponding parts are correctly matched.

Example: Consider two triangles, $\triangle ABC$ and $\triangle XYZ$. If $AB = XY$, $BC = YZ$, and $\angle B = \angle Y$, then by SAS congruence, $\triangle ABC \cong \triangle XYZ$. This means $\angle A = \angle X$, $\angle C = \angle Z$, and $AC = XZ$.

Similarity of Triangles

Two triangles are said to be similar if they have the same shape but not necessarily the same size. This means that all their corresponding angles are equal, and the ratios of their corresponding sides are equal. If triangle ABC is similar to triangle PQR, we denote it as $\triangle ABC \sim \triangle PQR$. This implies that:

Angle A = Angle P
Angle B = Angle Q
Angle C = Angle R
$\frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{RP}$

Similar triangles are like scaled versions of each other.

Similarity Criteria:

Similar to congruence, there are specific criteria to prove that two triangles are similar.

1. AA (Angle-Angle) Similarity Criterion

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. This is the most commonly used criterion because if two angles are equal, the third angle must also be equal (sum of angles = 180°).
If $\angle B = \angle Q$ and $\angle C = \angle R$, then $\triangle ABC \sim \triangle PQR$.

2. SSS (Side-Side-Side) Similarity Criterion

If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the two triangles are similar.
If $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{RP}$, then $\triangle ABC \sim \triangle PQR$.

3. SAS (Side-Angle-Side) Similarity Criterion

If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, then the two triangles are similar.
If $\angle B = \angle Q$ and $\frac{AB}{PQ} = \frac{BC}{QR}$, then $\triangle ABC \sim \triangle PQR$.

Key Distinction: Congruence vs. Similarity

Congruent triangles are identical in size and shape. All corresponding sides and angles are equal.
Similar triangles have the same shape (equal corresponding angles) but can have different sizes (proportional corresponding sides).

If two triangles are congruent, they are also similar. However, if two triangles are similar, they are not necessarily congruent.

Properties of Similar Triangles:

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides (or corresponding altitudes, or corresponding medians, or corresponding angle bisectors).
If $\triangle ABC \sim \triangle PQR$, then: $$\frac{Area(\triangle ABC)}{Area(\triangle PQR)} = \left(\frac{AB}{PQ}\right)^2 = \left(\frac{BC}{QR}\right)^2 = \left(\frac{CA}{RP}\right)^2$$ Let $h_1$ and $h_2$ be corresponding altitudes, then: $$\frac{Area(\triangle ABC)}{Area(\triangle PQR)} = \left(\frac{h_1}{h_2}\right)^2$$

Example: Consider $\triangle ABC$ and $\triangle XYZ$. If $\angle A = 50^\circ, \angle B = 60^\circ, \angle C = 70^\circ$ and $\angle X = 50^\circ, \angle Y = 60^\circ, \angle Z = 70^\circ$. By the AA criterion (or AAA), $\triangle ABC \sim \triangle XYZ$. If, additionally, $AB=10, BC=12, AC=14$ and $XY=5, YZ=6, XZ=7$, then the ratio of corresponding sides is $\frac{10}{5} = \frac{12}{6} = \frac{14}{7} = 2$. The triangles are similar with a scale factor of 2.

Pythagorean Theorem and Similarity: In a right-angled triangle, the altitude drawn from the vertex of the right angle to the hypotenuse divides the triangle into two smaller triangles that are similar to the original triangle and to each other. This property is fundamental to proving the Pythagorean theorem.

Application in Exams: Problems involving similar triangles often appear in geometry sections. Look for parallel lines (which create equal angles), shared angles, or triangles within larger triangles. The ratio of sides, perimeters, and areas are common questions. Always write down the similarity statement ($\triangle ABC \sim \triangle PQR$) carefully to match corresponding vertices.