Isosceles Triangles
Isosceles triangles are those triangles that have two sides of equal measure, while the third one is of different measure. We know that triangles are three-sided enclosed polygons and they are classified as equilateral, isosceles, or scalene, based on the length of their sides. In this section, we will learn about the definition of an isosceles triangle and its properties.
| 1. | What is an Isosceles Triangle? |
| 2. | Properties of Isosceles Triangle |
| 3. | Isosceles Triangle Theorem |
| 4. | Parallel Lines Symbol |
What is an Isosceles Triangle?
An isosceles triangle is a triangle that has two sides of equal length. Let us do a small activity to understand this better. Take a rectangular sheet of paper and fold it in half. Draw a line from the top folded corner to the bottom edge. You can see a triangle when you open the sheet. Mark the vertices of the triangles as A, B, and C. Now measure AB and AC. Repeat this activity with different measures and observe the pattern. We can observe that AB and AC are always equal. This type of triangle where two sides are equal is called an isosceles triangle.
Properties of Isosceles Triangle
Each geometric figure has some properties that make it different and unique from the others. Here is a list of a few properties of isosceles triangles:
- The two equal sides of an isosceles triangle are called the legs and the angle between them is called the vertex angle or apex angle.
- The side opposite the vertex angle is called the base and base angles are equal.
- The perpendicular from the vertex angle bisects the base and the vertex angle.
Isosceles Triangle Theorem
In an isosceles triangle, if two sides of a triangle are congruent, then angles opposite to those sides are congruent. Conversely, if the two angles of a triangle are congruent, the corresponding sides are also congruent. So we can say that ∠ABC = ∠ACB, and AB = AC in the given figure:
Now, we also know that in an isosceles triangle, the altitude from the apex angle (perpendicular) bisects the base and the apex angle. So, we can say that ∠BAD = ∠DAC, and BD = DC. With this information, we can say that ΔADB and ΔCDB are congruent.
Isosceles Triangle Area Formula
The area of an isosceles triangle can be calculated in many ways based on its known elements. Here is a list of formulas used to find the area of an isosceles triangle based on the given parameters.
| Known parameters | Area of an Isosceles triangle formula |
|---|---|
| When the base \(b\) and height \(h\) are known | 1/2 × b × h |
| When all the sides \( a\) and the base \(b\) are known | \[\frac{b}{2}\sqrt{\text{a}^2 - \frac{b^2}{4}}\] |
| When the length of the two sides \(a\) and \(b\) and the angle between them \(\angle \text{α}\) is known | \[\frac{1}{2} ab\:sin(\text{α}) \] |
Important Topics
Given below is the list of topics that are closely connected to the
Solved Examples on Isosceles Triangle
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Example 1: In the given triangle, find the measure of BD and area of triangle ADB.
Solution:
In an isosceles triangle, the perpendicular from the vertex angle bisects the base. So, BD = DC = 3 cm. In triangle ADB, base (b)= 3 cm, height (h) = 4 cm. The formula for the area of a triangle is 1/2 × b × h. Therefore, the area of the triangle ADB is 1/2 × 3 × 4 = 3 × 2 = 6 cm² .
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Example 2: Find the perimeter of an isosceles triangle, if the base is 24 cm and the area is 60 cm2.
Solution:
We know that the base is 24 cm and the area is 60 cm2 for the given triangle. We know that the formula for the area of a triangle is A = 1/2 × b × h. Substituting the values, we get 60 = 1/2 × 24 × h. Solving this equation, we get the height (h) = 5 cm.
Now, if we draw a perpendicular (height) from the vertex angle C to the base AB, we will be able to find the length of the equal sides (CA and CB). Applying Pythagoras theorem to the half of the triangle (including half of the base side and one leg of an equal side), we will now get the length of the equal sides of the given triangle. We get: √(52 + 122) = √169 = 13. Therefore, the equal sides (CA and CB) of the triangle ABC are 13 cm each. Now, Perimeter(p) = 2a + b = 2(13) + 24 = 50 cm.
FAQs on Isosceles Triangle
What are the Angles in an Isosceles Triangle?
An isosceles triangle has a vertex angle and two base angles. The base angles of an isosceles triangle measure the same.
What is an Isosceles Triangle?
A triangle in which two sides are equal is called an isosceles triangle. Following this fact, if two sides of a triangle are equal, then the angles opposite to those sides are also equal.
Explain the Isosceles Triangle Property.
The isosceles triangle property states that when two sides are equal, the base angles are also equal, and the perpendicular from the apex angle bisects the base.
How do you know if a Triangle is Isosceles?
A triangle can be scalene, isosceles, or equilateral when classified on the basis of the length of its sides. In a triangle, if any two sides are of equal length, it is considered to be an isosceles triangle.
Do Isosceles Triangles Add Up to 180?
The sum of the internal angles of all triangles is always 180°. Therefore, the angles of an isosceles triangle add up to 180°.
Which Triangle is a Right Isosceles Triangle?
In a right isosceles triangle, the equal sides form the right angle. In other words, any triangle with angles as 90°, 45°, 45° is a right isosceles triangle.
Can Isosceles Triangles be Right?
Yes, isosceles triangles can be right triangles if their three angles are 90°, 45°, and 45° respectively. In a right isosceles triangle, the equal sides join to form the right angle.