What are the important properties and formulas of an isosceles triangle?
The concept of properties of isosceles triangle plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. This topic helps you quickly identify triangle types, solve for missing angles or sides, and build a strong foundation for advanced geometry problems.
What Is Properties of Isosceles Triangle?
An isosceles triangle is a triangle that has exactly two sides of equal length. The angles opposite those two equal sides are also equal. You’ll find this concept applied in areas such as triangle classification, symmetry properties, and solving geometry questions involving medians, altitudes, and perimeters.
Key Properties of Isosceles Triangle
- The isosceles triangle has exactly two equal sides.
- The base angles (angles opposite the equal sides) are equal in measure.
- The vertex angle (between the equal sides) is usually different from the base angles.
- The altitude from the vertex to the base bisects both the base and the vertex angle.
- The altitude, median, and angle bisector from the vertex all lie on the same line.
- The isosceles triangle is symmetrical about the altitude from the vertex.
- The sum of all three angles is always 180° (angle sum property).
- An isosceles triangle can also be right angled or obtuse angled, but still must have two equal sides.
Key Formula for Properties of Isosceles Triangle
Here’s the standard formula for the area of an isosceles triangle when equal sides = a and base = b:
\[
\text{Area} = \frac{b}{4}\sqrt{4a^2 - b^2}
\]
For perimeter:
\[
\text{Perimeter} = 2a + b
\]
Step-by-Step Illustration
- Suppose in ΔABC, AB = AC and base BC = 6 cm, sides AB = AC = 5 cm.
The two equal sides are identified as “legs”, and the base is the different side. - To find area:
Use the formula:
Area = (b/4) × √(4a² − b²)
Plug in: b = 6, a = 5
= 6/4 × √(4×5² − 6²)
= 1.5 × √(100 − 36)
= 1.5 × 8 = 12cm²
- To find the vertex angle when base angles are known:
If base angles = 75°, find vertex angle:
Sum of angles = 180°,
Thus, vertex angle = 180° − 2×75° = 30°
Speed Trick or Vedic Shortcut
Here’s a quick trick to find the missing angle in an isosceles triangle: If you know the base angle (let’s call it x), then the vertex angle is always 180° − 2x. This saves time in exams.
Example Trick: If each base angle is 63°, vertex angle = 180 − 2×63 = 54°.
Tricks like this are very practical for competitions and school tests. Vedantu’s live classes often teach such shortcuts to boost accuracy.
Solved Example: Find Area and Perimeter
Given: Sides AB = AC = 10 cm, base BC = 12 cm.
1. Area:
Area = (b/4) × √(4a² − b²)
= (12/4) × √(4×100 − 144)
= 3 × √(400 − 144)
= 3 × √256
= 3 × 16 = 48cm²
2. Perimeter:
Perimeter = 2 × 10 + 12 = 32 cm
Try These Yourself
- Find the vertex angle if the base angles are each 68°.
- Calculate the area when equal sides = 13 cm, base = 10 cm.
- Which sides are congruent if ∆XYZ is isosceles and ∠X = ∠Y?
- Identify if a triangle with sides 7 cm, 7 cm, 10 cm is isosceles, scalene or equilateral.
Frequent Errors and Misunderstandings
- Confusing isosceles with equilateral (where all sides and angles are equal).
- Forgetting that the base can be the longest or shortest side.
- Assuming all isosceles triangles are right-angled—only those with 90°, 45°, 45° are.
- Miscalculating area by wrong substitution.
Comparison Table: Isosceles, Equilateral, Scalene
| Property | Isosceles Triangle | Equilateral Triangle | Scalene Triangle |
|---|---|---|---|
| Equal Sides | 2 | 3 | 0 |
| Equal Angles | 2 | 3 | 0 |
| Symmetry | 1 axis | 3 axes | None |
| Example | 5, 5, 8 | 6, 6, 6 | 4, 5, 6 |
Real-Life and Exam Usage
The properties of isosceles triangle appear in many real-world contexts: the design of bridges, flags, engineering beams, and even art. In Maths Olympiads and board exams, isosceles triangles commonly feature in geometry sections, MCQs, and proofs. You’ll see their properties used in coordinate geometry, constructions, and symmetrical patterns. Learning this well can improve your chances in competitive exams.
Relation to Other Concepts
The idea of isosceles triangle connects closely to topics like medians and altitudes, angle sum properties, and similarity of triangles. Mastering these basics helps with more advanced triangle properties in later grades.
Classroom Tip
A quick way to spot the isosceles triangle is to look for two sides of equal length or two matching markings in diagrams. Draw and fold paper triangles to visually check symmetry—a method Vedantu’s teachers use often in live class demonstrations to make learning interactive.
We explored properties of isosceles triangle—definition, key formula, quick tricks, solved examples, common errors, and how this connects to other important triangle topics. Keep reviewing and practicing with Vedantu’s study materials and triangle property lessons to become confident in geometry.
Related Vedantu Pages
FAQs on Properties of Isosceles Triangle Explained with Examples
1. What is an isosceles triangle?
An isosceles triangle is a triangle with at least two sides of equal length. These equal sides are called the legs, and the angle between them is the vertex angle or apex angle. The side opposite the vertex angle is the base.
2. What are the properties of an isosceles triangle?
Key properties of an isosceles triangle include:
- Two sides are equal in length.
- The angles opposite the equal sides (base angles) are equal in measure.
- The altitude from the vertex angle bisects the base and the vertex angle.
- The median from the vertex angle bisects the base.
- The altitude, median, and angle bisector from the vertex angle are all the same line segment.
3. How do you prove the base angles of an isosceles triangle are equal?
The proof involves constructing an altitude from the vertex angle to the base. This creates two congruent right-angled triangles. Since corresponding parts of congruent triangles are equal (CPCTC), the base angles are proven equal.
4. What is the formula for the area of an isosceles triangle?
The area (A) can be calculated using the formula: A = (1/2) * base * height, where 'base' is the length of the unequal side and 'height' is the perpendicular distance from the vertex angle to the base. Alternatively, if you know the lengths of all three sides (a, a, b), you can use Heron's formula.
5. Can an isosceles triangle be a right-angled triangle?
Yes, a right-angled isosceles triangle is possible. It has angles measuring 45°, 45°, and 90°.
6. What is the difference between an isosceles and an equilateral triangle?
An isosceles triangle has at least two equal sides, while an equilateral triangle has all three sides equal. An equilateral triangle is a special case of an isosceles triangle.
7. How do you find the perimeter of an isosceles triangle?
The perimeter (P) is the sum of all three sides. If the equal sides have length 'a' and the base has length 'b', then P = 2a + b.
8. What is the relationship between the altitude and median from the vertex angle in an isosceles triangle?
In an isosceles triangle, the altitude (height) from the vertex angle to the base is also the median (it bisects the base) and the angle bisector of the vertex angle.
9. Can an isosceles triangle be obtuse?
Yes, an isosceles triangle can be obtuse. In this case, the vertex angle will be greater than 90°, and the two base angles will be acute and equal.
10. How are isosceles triangle properties used in solving geometry problems?
Knowing that the base angles are equal allows you to set up equations to find unknown angles or sides. The properties of the altitude, median, and angle bisector being concurrent often help in proving congruence or finding lengths.
11. What is the formula to find the height of an isosceles triangle?
If you know the lengths of the equal sides (a) and the base (b), the height (h) can be found using the Pythagorean theorem: h = √(a² - (b/2)²).
12. Are all isosceles triangles similar?
No. Two isosceles triangles are similar only if their base angles are equal. Isosceles triangles with different base angles are not similar.
