Properties of Isosceles Triangle
Triangles are classified into different types on the basis of their sides and angles. Each of them has their own individual properties.
Classification of Triangles on the Basis of their Sides is as Follows:
Equilateral triangle: A triangle whose all the three sides are equal.
Isosceles triangle: A triangle whose two sides are equal.
Scalene triangle: A triangle whose all three sides are unequal.
Classification of Triangles on the Basis of their Angles is as Follows
Acute angled triangle: A triangle whose all interior angles are less than 900.
Right angled triangle: A triangle whose one interior angle is 900.
Obtuse angled triangle: A triangle whose one interior angle is more than 900.
Now, we will discuss the properties of an isosceles triangle.
An Isosceles Triangle has the Following Properties:
It has two sides of equal length. In the above figure, sides AB and AC are of equal length ‘a’ unit.
The angles opposite to equal sides are equal in measure. In the above figure, ∠ B and ∠C are of equal measure.
The altitude from vertex A to the base BC is the perpendicular bisector of the base BC.
The altitude from vertex A to the base BC is the angle bisector of the vertex angle ∠ A.
The altitude from the vertex divides an isosceles triangle into two congruent right-angled triangles. In the above figure, triangle ADB and triangle ADC are congruent right-angled triangles.
The median drawn from vertex A will bisect BC at right angles.
The medians drawn from vertex B and vertex C will not bisect the opposite sides AB and AC.
Two isosceles triangles are always similar.
In the right angled isosceles triangle, one angle is a right angle (90 degrees) and the other two angles are both 45 degrees.
In the right angled isosceles triangle, the altitude on the hypotenuse is half the length of the hypotenuse.
In the right angled isosceles triangle, the center of the circumcircle lies on the hypotenuse and the radius of the circumcircle is half the length of the hypotenuse.
An isosceles triangle whose two equal sides length is ‘a’ unit and length of its base is ’b’ unit. Then,
Area of isosceles triangle =
\[\frac{b}{2}\sqrt{a^{2} - \frac{b^{2}}{4}}\]
Perimeter of isosceles triangle = 2a + b
Solved Examples on Isosceles Triangles
1. Find the value of x in the given isosceles triangle ABC.
Solution: In the given isosceles ∆ABC, AB = AC.
Therefore, ∠ABC = ∠ACB = 75°. (Angle opposite to equal sides are equal)
Now, ∠ABC + ∠BAC + ∠ACB = 180°. (Interior angle sum property of a triangle)
⟹ 75° + x° + 75° = 180°
⟹ x° = 180° - 150°
⟹ x° = 30°
2. Find the value of x in the given isosceles triangle PQR, whose base QR is extended to S.
In the given isosceles triangle PQR, PQ = PR.
Therefore, ∠PQR = ∠PRQ = x°. (Angle opposite to equal sides are equal)
Now, ∠PRQ + ∠PRS = 180°. (By linear pair)
⟹ x° + 120° = 180°.
⟹ x° = 180° - 120°.
⟹ x° = 60°.
3. Find the perimeter and area of an isosceles triangle whose two equal sides and base length is 5 cm and 6 cm respectively.
Solution: Given, length of two equal sides of an isosceles triangle = a = 5 cm
And length of its base = b = 6 cm
Perimeter of an isosceles triangle = 2a + b
= 2(5) + 6
= 10 + 6 = 16 cm.
We know that the area of an isosceles triangle is \[\frac{b}{2}\sqrt{a^{2} - \frac{b^{2}}{4}}\], now putting the values of a and b, we have:
=\[\frac{6}{2}\sqrt{5^{2} - \frac{6^{2}}{4}}\]
=\[ 3\sqrt{36 -9}\]
= \[3 \times \sqrt{16}\]
= \[3 \times 4 = 12cm^{2}\]
4. If two angles of a triangle are 60° each, then the triangle is:
(a) Isosceles but not equilateral
(b) Scalene
(c) Equilateral
(d) Right-angled
Solution: (c)
By interior angle sum property of triangle,
The measure of the third angle of the given triangle comes out to be 60°.
So, each interior angle of a given triangle is 60°, which means each side of the triangle is equal (the sides opposite to equal angles are equal). Thus. The triangle is an equilateral triangle.
5. If in an isosceles triangle, each of the base angles is 40°, then the triangle is:
(a) Right-angled triangle
(b) Acute angled triangle
(c) Obtuse angled triangle
(d) Isosceles right-angled triangle
Solution: (c)
By interior angle sum property of triangle,
The measure of the third angle of the given triangle comes out to be 120°.
Since, the measure of one of the angles of the triangle is greater than 90°. Therefore, the given triangle is an obtuse angled triangle.
In this article we will discuss the meaning of the isosceles triangle, its properties and formula and if you solved examples. The frequently asked questions at the end of the article can help you If you encounter any doubts while reading the same.
Few More Properties of Isosceles Triangle
The properties of an Isosceles Triangle are:
The two sides of an Isosceles Triangle are congruent.
Since the two sides are congruent, therefore, the two angles made by the two equal sides with the base are also congruent.
The altitude of an Isosceles Triangle divides the triangle into two equal right-angled triangles.
The altitude of an Isosceles Triangle divides the base into two equal parts.
Height and Area of an Isosceles Triangle Formula
Height of an Isosceles Triangle
If the two equal sides of the Isosceles Triangle as shown in the figure are denoted by the letter aa and, bb is the base of the triangle, then the height hh is given by:
\[h = \frac{1}{2}\sqrt{4a^{2} - b^{2}}\]
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\[h = \sqrt{a^{2} - \frac{b^{2}}{4}}\]
Area of Isosceles Triangle using Base and Height
If the Base (b)(b) and the Height (h)(h) of an Isosceles Triangle as shown in the figure above are known, then the Isosceles Triangle Formula for finding the area is given by:
A= 1/2×b×h
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Perimeter of Isosceles Triangle
The Perimeter of the Isosceles Triangle may be defined as the sum of the length of all three sides. Therefore, The Perimeter of the Isosceles Triangle having two equal sides aa and base bb is given by,
p=2a+b
Miscellaneous Examples on Isosceles Triangles
1. The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base angle?
Every triangle has 180 degrees. So, Let x = the vertex angle and 2x+5 = the base angle
So the equation to solve becomes x+(2x+5)+(2x+5)=180
Therefore,
Vertex angle = 34 and base angle= 73.
2. What is the area of an isosceles triangle with base b of 6 cm and lateral a side 5 cm?
Apply Pythagoras theorem:
\[a^{2} = (\frac{b}{2})^{2} + h^{2}\]
Area A = (1/2) * b * h = (1/2) 3 * 3 = 9 unit²
FAQs on Isosceles Triangle
1. What is Pythagoras theorem?
The Pythagorean Theorem states that in any right triangle, the sum of the squares of the lengths of the triangle’s legs is the same as the square of the length of the triangle’s hypotenuse. This theorem is represented by the formula a²+b²=c². You can simply use the Pythagorean theorem to find the length of the third side.
2. Are all sides of an isosceles triangle equal?
A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is called a scalene triangle. An equilateral triangle has all the three sides and angles equal. It is a special case of an isosceles triangle including the isosceles right Triangle.
3. What are some of the Other types of triangles?
Some of the other types of triangles include:
Right Triangle : which has a one right angle that is equal to 90 degrees.
Equilateral triangle: all the sides are the same in an equilateral triangle.
Obtuse Triangle: It is a triangle in which one angle is greater than 90 degrees and less than 180 degrees. For example, if one angle of a triangle is 120 degrees and other two angles are 30 degrees each, it is said to be an obtuse triangle.
Scalene Triangle: in which no sides have the same length.
Acute Triangle : It is a triangle in which all angles are less than 90 degrees. For example, if all angles in a triangle are equal to 60 degrees, it is an acute triangle.
4. If an isosceles triangle has an angle measuring greater than 100 degrees, and another angle with a measuring z degree, which of the following is true?
In order for a triangle to be an isosceles triangle, it must contain two equivalent angles and one angle that is different. Given that one angle is greater than 100 degrees: 180−100=80degrees.The sum must be less than 80°.If an angle is represented by z:
2z<80degrees=z<40degrees.
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