TRIANGLES
Frequently Asked Questions
A triangle is a closed figure formed by three line segments and has three vertices, three sides, and three angles.
When two triangles have the same shape and size, their corresponding sides and angles are equal; they are said to be congruent.
The main congruence rules are SSS, SAS, ASA, AAS, and RHS for right triangles.
Two triangles are similar if their corresponding angles are equal and corresponding sides are in proportion.
AAA / AA, SAS similarity, and SSS similarity.
If two angles of one triangle are equal to two angles of another, the triangles are similar.
If a line is drawn parallel to one side of a triangle to intersect the other two sides, it divides the sides proportionally.
Thales’ Theorem is another name for the Basic Proportionality Theorem (BPT).
If a line divides any two sides of a triangle in the same ratio, the line must be parallel to the third side.
In a right-angled triangle: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
If for a triangle \(a^2 + b^2 = c^2\), the triangle is right-angled.
The ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
The sides and angles that occupy the same relative position in congruent or similar triangles.
By showing the ratio of all three pairs of corresponding sides is equal.
If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and one side of another right-angled triangle, the triangles are congruent.
Used in map-making, architecture, engineering, shadow measurement, surveying, and scaling models.
The ratio of corresponding sides of similar triangles.
Corresponding sides are in proportion.
\(\frac{Area_1}{Area_2} = \left(\frac{side_1}{side_2}\right)^2\).
Perimeters are in the same ratio as corresponding sides.
Yes. Similarity requires same shape, not same size.
It means the ratios of corresponding sides are equal.
The line joining midpoints of two sides of a triangle is parallel to the third side and half of it.
It is a specific case of BPT where each side is divided in the ratio 1:1.
Helps in dividing lines proportionally and constructing parallel segments.
Look at the relative position of vertices in both triangles.
Being at equal distance from two or more points/lines.
A triangle with one angle equal to \(90^\circ\).
The longest side is hypotenuse; the other two are legs or perpendicular and base.
Trigonometric ratios are defined based on similar right triangles, making ratios consistent.
Helps to calculate the distance between two points.
Yes, but triangles are simplest because if two triangles have two equal angles, the third automatically matches.
A geometric tool used to divide lengths in fixed ratios, often based on triangle similarity.
Used in navigation, construction, height-distance problems, physics, and engineering.
By drawing a line parallel to one side, forming two similar triangles, then equating ratios of corresponding sides.
Dividing land plots proportionally using parallel boundaries.
Operations such as scaling, rotation, reflection, and translation, which preserve similarity.
A transformation that enlarges or reduces a figure proportionally—basis of similarity.
BPT (Thales’), its converse, Similarity criteria, Area ratio theorem, Pythagoras theorem with converse.
Typically 4–8 marks in CBSE Class 10, including one theorem-based proof question.
Check for equal angles first; then verify proportional sides.
Yes, because all angles are \(60^\circ\) and sides are proportional.
No, only if the angles also match.
Three positive integers \(a, b, c\) satisfying \(a^2 + b^2 = c^2\), e.g., (3,4,5).
Allow quick checking of right triangles without calculation.
Yes, all corresponding angles remain equal.
No, areas change by the square of scale factor.
If \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{a+c}{b+d}\) is also an intermediate proportional ratio.
The point lies between the endpoints of the segment.
Not necessarily; their shapes may differ.
The sides must satisfy Pythagorean equality.
Proofs, ratio-based calculations, similarity applications, height-distance applications, MCQs.
Identify corresponding sides \(\Rightarrow\) set ratios \(\Rightarrow\) cross-multiply \(\Rightarrow\) solve.
By forming two similar triangles using shadows, poles, or angles of elevation.
Two right triangles with one equal acute angle.
Use the equation \(\frac{side_1}{side_2} = \text{scale factor}\).
Using wrong pair of sides for ratio; ratio must correspond to the intersected sides.
Yes, it deals with division of sides in a triangle using parallel lines.
Show equality of sides/angles using SSS/SAS/ASA/RHS.
Maps are reduced versions (scaled diagrams) using constant ratios.
AAA ensures same shape but not same size, so not congruence.
Drawing additional lines to help prove similarity/congruence.
Cameras (lens projection), GPS, surveying instruments, and theodolites.
To determine distances and elevations indirectly.
Sometimes median-based constructions produce smaller similar triangles.
Yes, orientation or position does not affect similarity.
All polygons can be divided into triangles, making them the building blocks of geometry.
No, they must also share one acute angle.
Check quickly whether the square of the longest side equals the sum of squares of other two.
Because it supports trigonometry, coordinate geometry, mensuration, and real-world calculations.
