How to Identify, Classify, and Solve Triangle Questions in Maths
A triangle is a polygon that consists of three sides having three vertices. It is a two-dimensional figure. The interior angle of the triangle is measured to be 180 degrees. It is a polygon that has the least number of sides.
Types of Triangles
Triangles are broadly classified based on the length of the sides and interior angles of the triangle.
Based on the sides, triangles are classified into three types, they are:
Scalene Triangle.
Isosceles Triangle.
Equilateral Triangle.
Scalene Triangle
A scalene triangle is a triangle where all three sides have different lengths. As the lengths of the sides are different, the interior angles are also different.
Isosceles Triangle
An isosceles triangle is a triangle where two out of three sides are equal. As the two sides are equal, the angles opposite to the equal sides are also equal.
Equilateral Triangle
An equilateral triangle is a triangle where all three lengths of the sides are equal. Such that, all the interior angles have an angle of 60 degrees since the sum of interior angles angle is 180 degrees. It is also referred to as an equiangular triangle.
Based on the interior angles, triangles are classified into three types, they are:
Acute angle triangle
Obtuse angle triangle
Right angle triangle
Acute Angle Triangle
A triangle where all three interior angles are less than 90 degrees is called an acute-angled triangle.
Obtuse Angle Triangle
A triangle that has one of the three interior angles greater than 90 degrees is called an obtuse-angled triangle.
Right Angle Triangle
A triangle where one of the three sides is exactly 90 degrees is called a right-angled triangle. The side opposite to this 90-degree angle has the longest side length, it is called the hypotenuse.
If any two interior angles of a triangle are similar to any two interior angles of another triangle, then the two triangles are said to be similar to each other.
For instance,
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Consider two triangles ΔABC and triangle ΔXYZ
If ∠ A = ∠X and ∠C = ∠Z
then, ΔABC ~ ΔXYZ
Therefore,
AB/XY = BC/YZ = AC/XZ and ∠B = ∠Y
SAS or Side-Angle-Side Similarity
The SAS Similarity Theorem defines that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent then that two triangles will be termed as similar.
SSS or Side-Side-Side Similarity
If all the three sides of a triangle are equal to the three sides of another triangle, then the two triangles are similar.
For instance,
Consider two triangles ΔABC and triangle ΔXYZ
If AB/XY = BC/YZ = AC/XZ,
then, ΔABC ~ ΔXYZ.
Therefore,
∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z
Solved Problems
1. If A ABC ~ ARPQ, AB = 3 cm, BC = 5 cm, AC = 6 cm, RP = 6 cm and PQ = 10 cm, then Find QR.
Solution: ΔABC ~ ΔRPQ
So, AB/RP = BC/PQ = AC/RQ as similar triangles have proportional sides
3/6=5/10=6/QR
½ = ½ 6/QR
QR= 12 cm
2. Find the Area of a Triangle Whose Two Sides are 12 cm and 16 cm and the Perimeter is 42cm.
Ans. Assume that the third side of the triangle is “x”.
Now, the three sides of the triangle are 12 cm, 16 cm, and “x” cm
It is given that the perimeter of the triangle = 36cm
So, x = 36 – (12 + 16) cm = 8 cm
∴ The semi perimeter of triangle = 36/2 = 18
Using Heron’s formula,
Area of the triangle,
\[= \sqrt{s(s-a)(s-b)(s-c)}\]
\[= \sqrt{18(18-12)(18-16)(18-8)}\] cm²
\[= \sqrt{18 \times 6 \times 2 \times 11}\]m²
\[= 18\sqrt{132}\] cm²
3. Corresponding Sides of Two Similar Triangles are in the Ratio of 1:2. If the Area of the Small Triangle is 16 sq. cm, then the Area of the Large Triangle is:
(a) 30 sq.cm.
(b) 6 sq.cm
(c) 7 sq.cm.
(d) 4 sq.cm
Answer: d
Solution: Let A1 and A2 are areas of the small and large triangle.
Then,
A2/A1=(side of large triangle/side of the small triangle)
A2/18=(1/2)2
A2=108 sq.cm.
4. Sides of Two Similar Triangles are in the Ratio 1: 9. Areas of these Triangles are in the Ratio
(a)2: 81
(b)1: 9
(c)1: 16
(d)16: 81
Answer: c
Explanation: Let ABC and DEF are two similar triangles, such that,
ΔABC ~ ΔDEF
And AB/DE = AC/DF = BC/EF = 1/9
As the ratio of the areas of these triangles will be equal to the square of the ratio of the corresponding sides,
∴ Area(ΔABC)/Area(ΔDEF) = AB2/DE2
∴ Area(ΔABC)/Area(ΔDEF) = (1/9)2 = 1/81 = 1: 81
Hence the above article is very useful as it provides solved examples that will help students to clear their doubts. Types of triangles are also explained in the article.
FAQs on Triangles Questions: Types, Properties & Exam Tips
1. What is the mathematical definition of a triangle?
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry, formed by connecting three non-collinear points. Its key characteristic is that it is a closed two-dimensional shape with three straight sides.
2. How are triangles classified based on the length of their sides?
Triangles are classified into three types based on the lengths of their sides:
- Equilateral Triangle: All three sides are of equal length, and consequently, all three interior angles are equal (60° each).
- Isosceles Triangle: At least two sides are of equal length. The angles opposite these equal sides are also equal.
- Scalene Triangle: All three sides have different lengths, and all three interior angles are also different.
3. How are triangles classified based on their interior angles?
Based on their interior angles, triangles are categorised as follows:
- Acute-angled Triangle: All three interior angles are acute, meaning each angle is less than 90°.
- Right-angled Triangle: One of the interior angles is exactly 90° (a right angle). The side opposite the right angle is called the hypotenuse.
- Obtuse-angled Triangle: One of the interior angles is obtuse, meaning it is greater than 90°.
4. What is the Angle-Angle (AA) similarity criterion for triangles?
The Angle-Angle (AA) similarity criterion, a key concept in the CBSE syllabus, states that if two angles of one triangle are congruent (equal in measure) to two angles of another triangle, then the two triangles are similar. This implies their third angles are also equal, and the ratio of their corresponding side lengths is constant. For example, if in ΔABC and ΔXYZ, ∠A = ∠X and ∠B = ∠Y, then ΔABC ~ ΔXYZ.
5. What are the fundamental properties every triangle has?
Every triangle, regardless of its type, shares some fundamental properties. The most important ones are:
- Angle Sum Property: The sum of the measures of the three interior angles is always 180 degrees.
- Exterior Angle Property: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two opposite interior angles.
- Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
6. Why is the sum of the angles in any triangle always 180 degrees?
This fundamental property can be proven using parallel lines. Imagine a triangle ABC. If you draw a line through vertex A that is parallel to the base BC, you create three angles on that straight line. The two outer angles are alternate interior angles to angles B and C, making them equal. The middle angle is angle A of the triangle. Since angles on a straight line add up to 180°, the sum of these three angles (which is equivalent to A + B + C) must also be 180 degrees.
7. What is the difference between congruent and similar triangles?
The main difference lies in what is being preserved.
- Congruent triangles have the exact same size and shape. This means all their corresponding sides and corresponding angles are equal. You can think of them as perfect copies of each other.
- Similar triangles have the same shape but can have different sizes. Their corresponding angles are equal, but their corresponding sides are in proportion. One is essentially a scaled-up or scaled-down version of the other.
8. Are there really 12 types of triangles? How are they named?
This is a common point of confusion. There are not 12 distinct types, but rather combinations of classifications. Triangles are primarily classified in two ways: by their sides (Equilateral, Isosceles, Scalene) and by their angles (Acute, Right, Obtuse). We often combine these names to describe a triangle more precisely, for example:
- An Acute Isosceles Triangle (two equal sides, all angles < 90°).
- A Right Scalene Triangle (one 90° angle, all sides different).
- An Obtuse Isosceles Triangle (one angle > 90°, two equal sides).
9. Can a triangle have two obtuse angles? Why or why not?
No, a triangle cannot have two obtuse angles. An obtuse angle is defined as an angle greater than 90°. If a triangle had two such angles, their sum alone would exceed 180° (e.g., 91° + 91° = 182°). This violates the fundamental Angle Sum Property, which states that the sum of all three interior angles of any triangle must be exactly 180°. Therefore, a triangle can have at most one obtuse angle.
10. Where are the properties of triangles used in real life?
The properties of triangles are crucial in many real-world applications. For instance:
- Architecture and Engineering: The rigid structure of triangles is used in bridges, roof trusses, and domes to distribute weight and provide stability.
- Navigation and GPS: The method of triangulation is used to determine locations by forming a triangle between two known points and the point to be located.
- Art and Design: Artists use triangles for perspective and composition to create a sense of depth and balance.
- Surveying: Land surveyors use trigonometry and triangle properties to measure distances and areas of land.
