Isosceles triangles are those triangles that have at least two sides of equal measure. We know that triangles are three-sided enclosed polygons and they are classified as equilateral, isosceles, and scalene, based on the length of their sides. In this article, 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 Angles|
|4.||Scalene Equilateral and Isosceles Triangle|
|5.||FAQs on Isosceles Triangle|
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 (as shown in the figure below). You can see a triangle when you open the sheet. Mark the vertices of the triangle as O, D, and C. Now measure OD and OC. Repeat this activity with different measures and observe the pattern. We can observe that OD and OC are always equal. This type of triangle where two sides are equal is called an isosceles triangle.
In the above figure, △ODC is an isosceles triangle with OD = OC and ∠ODC = ∠OCD. Let us now learn some of the properties of isosceles triangles in the section below.
Properties of Isosceles Triangle
Each geometric shape has some properties that make it different and unique from the others. Here is a list of a few properties of isosceles triangles:
- Two equal sides and two equal angles.
- 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 apex angle bisects the base and the apex angle.
- The perpendicular drawn from the apex angle divides the isosceles triangle into two congruent triangles and is also known as its line of symmetry.
Isosceles Triangle Angles
Like any other triangle, there are three angles in an isosceles triangle which adds up to 180 degrees. Out of the three interior angles, the angles apart from the apex angle are equal in measure. The isosceles triangle theorem states that the angles opposite to the equal sides of an isosceles triangle are equal in measurement. So, in an isosceles triangle △ABC where AB = AC, we have ∠B = ∠C.
If the measure of the equal angles is less than 45 degrees each, then the apex angle will be an obtuse angle. If each of the equal angles measures exactly 45 degrees, then the apex angle will be a right angle. And, if each of the equal angles measures more than 45 degrees and less than 90 degrees, the apex angle will be an acute angle.
Scalene Equilateral and Isosceles Triangle
The three common types of triangles are scalene, equilateral, and isosceles triangles. Each triangle is different from the other on the basis of its unique properties. A scalene triangle is the one in which all three sides and all three angles are of different measurements, an equilateral triangle is the one with all three sides and angles equal, and in an isosceles triangle, two sides and two angles are equal in measurement. Look at the table below to understand the differences and similarities in scalene, equilateral, and isosceles triangles.
|Criteria||Scalene Triangle||Isosceles Triangle||Equilateral Triangle|
|Sides||All three sides are of different measurements.||At least two sides are equal in measure.||All three sides are equal in length.|
|Angles||All three interior angles are different.||At least two angles are equal in measure.||All three angles are equal and measure 60 degrees each.|
|Perpendicular Bisector||No specific relation||The perpendicular bisector drawn from the apex angle bisects that angle and the unequal side of the triangle.||The perpendicular bisector drawn from any angle bisects that angle and the side opposite to it.|
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Isosceles Triangle Examples
Example 1: In the given triangle, find the measure of BD and area of triangle ADB.
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.
Example 2: Find the perimeter of an isosceles triangle, if the base is 24 inches and the equal sides are 36 inches each.
We know that formula of the perimeter of an isosceles triangle (p) = 2a + b, where a is the length of each of the equal sides.
Here, a = 36 inches and b = 24 inches.
Substituting the values in perimeter of an isosceles triangle formula, we get, P = 2(36) + 24 = 96 inches.
Hence, the perimeter of the given triangle is 96 inches.
Example 3: What is the area of an isosceles triangle with a base of 8 units and lateral side 5 units?
We know that formula of the area of an isosceles triangle (A) = b/4[√(4a2 - b2)], where a is the length of each of the equal sides.
Here, a = 5 units and b = 8 units.
Substituting the values in the isosceles triangle area formula, we get, A = 2√(4 × 25 - 64) = 12 square units.
Hence, the area of the given triangle is 12 square units.
FAQs on Isosceles Triangle
What is an Isosceles Triangle?
A triangle in which at least 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.
What is the Isosceles Triangle Theorem?
The isosceles triangle theorem states that when two sides are equal, the base angles are also equal. The converse of isosceles triangle theorem is also true which states that in a triangle if two angles are equal then the sides opposite to those angles are also equal.
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 have Equal Angles?
In an isosceles triangle, two angles are equal in measure. These angles lie opposite to the equal sides. When all three angles are equal, it is known as an equilateral 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.
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. It contains the properties of both right triangles and isosceles triangles.
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 and the hypotenuse is the unequal side.
How to Find the Area of an Isosceles Triangle?
The area of an isosceles triangle can be derived by using Heron's formula: Area (A) = b/4[√(4a2 - b2)], where a is the length of the equal side and b is the base of the triangle. When the base and height/altitude of the triangle are given, then the isosceles triangle area can be found by using the formula A = 1/2 × base (b) × height (h) square units.
What are the Properties of an Isosceles Triangle?
A few important properties of an isosceles triangle are listed below:
- At least two sides are equal in length.
- The angles opposite to the equal sides are equal in measure.
- The perpendicular drawn from the vertex angle of an isosceles triangle acts as a line of symmetry that divides the triangle into two congruent triangles.
- The perpendicular drawn from the vertex angle bisect that angle and the side opposite to it.
- The area of an isosceles triangle using the side lengths can be calculated by using the formula Area (A) = b/4[√(4a2 - b2)] square units, where a = length of the equal side, and b = base of the triangle.
- The perimeter of the isosceles triangle is equal to 2a + b units, where a = length of the equal side, and b = base of the triangle.
What is the Perimeter of Isosceles Triangle?
The perimeter of any triangle is the sum of all its three sides. The isosceles triangle perimeter can be found by using the formula P = 2a + b units, where b is the base and a is the length of the equal side.
What is the Vertex Angle of an Isosceles Triangle?
The vertex angle of an isosceles triangle is the angle other than the two equal sides. It connects the two equal sides of that triangle. The vertex angle is also known as the apex angle of the triangle.