Theorems and Related Questions in Isosceles Triangle: A triangle is a polygon with three sides. We can categorise triangles based on their sides and angles. Triangles can be classified into three types based on their sides: equilateral, isosceles, and scalene triangles. A triangle with two equal sides is known as an isosceles triangle.

An isosceles triangle can be a right-angled triangle if one of the angles is a right angle. Isosceles triangles have some specified characteristics. In this article, we will learn theorems and proofs associated with isosceles triangles and solve some questions related to the theorems.

Isosceles Triangle

If any two of the three sides of a triangle are equal to each other, then the triangle is called an isosceles triangle.

In the above triangle, the two equal sides are indicated. It is an isosceles triangle. In an isosceles triangle, the two angles opposite to the two equal sides are congruent.

Properties of Isosceles Triangle

Each geometric figure has some specific features that make it unique from the others. Here, a few properties of isosceles triangles are given.

1. It has two identical or equal sides. 
2. It has two equal base angles.
3. The angle between two equal sides is the vertex angle, and the side opposite to the vertex angle is the triangle’s base. The perpendicular drawn from the vertex angle bisects the base and the vertex angle.
4. The altitude and the median are the same for an isosceles triangle.
5. If base angles are equal to \(45\) degrees and vertex angle (angle other than base angle) is equal to right angle \(\left(90^{\circ}\right)\), then the given isosceles triangle is called a right-angled isosceles triangle. One interesting fact is that an equilateral triangle is also an isosceles triangle (special case).

Isosceles Triangle Theorems

Theorem 1: Angles opposite to the equal sides of an isosceles triangle are also equal.

To prove: Consider an isosceles \(\Delta ABC\) where \(AC=BC\) ( as it has only two sides equal).

Here, we must prove that the angles opposite the sides \(AC\) and \(BC\) are equal, that is, \(\angle C A B=\angle C B A\)

Construction: Let us draw a bisector of \(\angle A C B\) and name it as \(CD\).

Now, we have \(\Delta ACD\) and \(\Delta BCD\)

In \(\Delta ACD\) and \(\Delta BCD\)

\(AC = BC\)      (given)                                                          

\(\angle A C D=\angle B C D\)   (By construction \(CD\) bisects \(\angle A C B)\)

\(CD=CD\) (Common sides in both triangles)

Therefore, \(\Delta ACD \cong \Delta BCD\) (By SAS congruence criterion)

So, \(\angle C A B=\angle C B A\)  (By CPCT)

Hence it is proved that angles opposite to equal sides are equal.

Theorem 2: Sides opposite to the equal angles of an isosceles triangle are equal.

To prove: In a \(\Delta ABC\), if the base angles are equal then, \(\Delta ABC\) is an isosceles triangle

Construction:  A perpendicular bisector \(CD\) is drawn on side \(AB.\)

Proof: Now, in \(\Delta ACD\) and \(\Delta BCD\) we have,

\(\angle A C D=\angle B C D\) (By construction)

\(CD=CD\) (Common sides in both triangles)

\(\angle A D C=\angle B D C=90^{\circ}\) (By construction \(CD\) is perpendicular to \((AB)\)

Therefore, \(\Delta ACD \cong \Delta BCD\) (By ASA congruence criterion)

So, \(AC=BC\) (By CPCT)

Hence, \(\Delta ABC\) is an isosceles triangle.

Altitude of an Isosceles Triangle

If any two of the three sides of a triangle are equal, the triangle is called an isosceles triangle.The altitude of an isosceles triangle bisects the angle of the vertex and bisects the base. Thus, the altitude of the isosceles triangle divides the triangle into two congruent triangles using SSS congruency.

The formula of altitude of an isosceles triangle, \(h=\sqrt{a^{2}-\frac{b^{2}}{4}}\)
Where \(a\) is the length of the congruent sides, and \(b\) is the base.

Formula of Perimeter of an Isosceles Triangle

If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an Isosceles triangle, its perimeter is given by \(=a+a+b=(2a+b)\) units.

Formula of Area of an Isosceles Triangle

If the length of the identical sides and the length of the base of an isosceles triangle are known, then the height or altitude of the triangle can be determined. The formula to calculate the area of an Isosceles triangle using sides is given as,

The formula of area of an isosceles triangle using only sides \(=\frac{1}{2} \times b \times \sqrt{a^{2}-\frac{b^{2}}{4}}\)  when, \(b=\) base of the isosceles triangle, \(h=\) height of the isosceles triangle, \(a=\) length of the two equal sides.

Solved Examples

Q.1. Calculate the perimeter of an isosceles triangle whose base is \(3\,{\rm{cm}},\) and congruent sides are \(5\,{\rm{cm}}.\)
Ans: Given that the congruent sides are \(5 \mathrm{~cm}\) and the base is \(3 \mathrm{~cm}\).
Let us consider the congruent sides are \(a\) and the base is \(b\).
If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an isosceles triangle, its perimeter is given by \(=a+a+b=(2a+b)\) units.
Now, the perimeter of the given triangle \(=(2 \times 5+3)=13 \mathrm{~cm}\)
Hence, the perimeter of the isosceles triangle is \(13 \mathrm{~cm}\)

Q.2. Find the height \(\left( h \right)\) of the triangle with the length of the sides is \(10\,{\rm{cm,}}\,{\rm{10}}\,{\rm{cm,}}\,{\rm{12}}\,{\rm{cm}}{\rm{.}}\)
Ans: Given that the length of the sides of the triangle is \(10 \mathrm{~cm}, 10 \mathrm{~cm}, 12 \mathrm{~cm}\)

So, \(a=10 \mathrm{~cm}\) and \(b=12 \mathrm{~cm}\)
We know that height of the isosceles triangle is \(\sqrt{\left(a^{2}-\frac{b^{2}}{4}\right)} .\)
\(=\sqrt{10^{2}-\frac{12^{2}}{4}}\)
\(=\sqrt{100-\frac{144}{4}}\)
\(=\sqrt{100-36}\)
\(=\sqrt{64}=8 \mathrm{~cm}\)
Hence, the height \((h)\) of the given triangle is \(8 \mathrm{~cm} .\)

Q.3. The base of an isosceles triangle is \(10\,{\rm{cm,}}\) and the perimeter is \(40\,{\rm{cm.}}\) Find the measure of the other two sides.
Ans: Given that the base is \(10 \mathrm{~cm}\) and the perimeter is \(40 \mathrm{~cm}\).
Let us say the congruent sides are \(a\) and the base is \(b\).
If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an isosceles triangle, its perimeter is given by \(=a+a+b=(2a+b)\) units.
Now, substituting \(b=10\) in the above we have, \((2a+b)=2a+10\)
According to the statement, \(2a+10=40\)
We will solve for \(a\) by transposing \(10\) from LHS to RHS.
So, we have, \(2 a=40-10=30 \mathrm{~cm}\)
Hence, \(a=\frac{30}{2}=15 \mathrm{~cm}\)
Therefore, the measure of the other two sides of the isosceles triangle is \(15 \mathrm{~cm} .\)

Q.4. Calculate the length of the altitude of an isosceles triangle whose base is \(3\,{\rm{cm}},\) and congruent sides are \(5\,{\rm{cm}}.\)
Ans: Given that the congruent sides are \(5 \mathrm{~cm}\), and the base is \(3 \mathrm{~cm}\).
Let us say the congruent sides are \(a\) and the base is \(b\).
Now, the formula of the altitude of the isosceles triangle \(=\sqrt{a^{2}-\frac{b^{2}}{4}}\)
\(=\sqrt{5^{2}-\frac{3^{2}}{4}}\)
\(=\frac{\sqrt{91}}{2} \mathrm{~cm}\)
Hence, the length of the altitude of the triangle is \(\frac{\sqrt{91}}{2} \mathrm{~cm}\)

Q.5. Find the triangle area with the length of the sides is \(3\,{\rm{cm,}}\,{\rm{3}}\,{\rm{cm,}}\,{\rm{4}}\,{\rm{cm}}{\rm{.}}\)
Ans: Given, length of the sides of the triangle is \(3 \mathrm{~cm}, 3 \mathrm{~cm}, 4 \mathrm{~cm} .\)

So, \(a=3 \mathrm{~cm}\) and \(b=4 \mathrm{~cm}\)
We know that area of the isosceles triangle is  \(\frac{1}{2} \times b \times \sqrt{\left(a^{2}-\frac{b^{2}}{4}\right)}\)
\(=\frac{1}{2} \times 4 \times \sqrt{3^{2}-\frac{4^{2}}{4}}\)
\(=2 \times \sqrt{9-4}\)
\(=2 \sqrt{5} \mathrm{~cm}^{2}\)
Hence, the area of the given triangle is \(2 \sqrt{5} \mathrm{~cm}^{2}\)

Summary

A triangle with two equal sides and two equal angles is known as an isosceles triangle. An isosceles triangle can be a right-angled triangle if one of the angles is a right angle. Isosceles triangles have some specified characteristics. This article explained the theorems and proofs related to isosceles triangles.

Theorem \(1\) states that the angles opposite to the equal sides of an isosceles triangle are also equal. Theorem \(2\) states that the sides opposite to the equal angles of a triangle are equal. It also explained the formulas related to the perimeter, altitude and area of an isosceles triangle. At last, it shows some solved examples related to the topic.

Frequently Asked Questions (FAQs)

Q.1. Write the theorems related to the isosceles triangles.
Ans: Theorem 1: Angles opposite to the equal sides of an isosceles triangle are also equal. 
Theorem 2: Sides opposite to the equal angles of a triangle are equal.

Q.2. Are a median and an altitude drawn on the base from opposite vertex angle the same for an isosceles triangle?
Ans: Yes, a median and an altitude of an isosceles triangle are the same if they are drawn from the vertex angle to the base opposite to it.

Q.3. What is the formula of the area of an isosceles triangle?
Ans: The general formula used to calculate the area of an isosceles triangle with base “\(b\)” and legs “\(a\)” is given by \(\frac{1}{2} \times b \times \sqrt{\left(a^{2}-\frac{b^{2}}{4}\right)} .\)

Q.4. Can an isosceles triangle be a right triangle? What will be its perimeter?
Ans: Yes, an isosceles triangle can be a right triangle if the vertex angle measures \(90^{\circ} .\) The perimeter of an isosceles right-angled triangle is obtained by finding the sum of the length of all sides. If the lengths of the other two sides are \(‘l’\) and the length of the hypotenuse is \(‘h’\) units, and then the perimeter of an isosceles right triangle is the sum of three sides.
Perimeter an isosceles right triangle \(= h + l + l\)

Q.5. How do you find the third side of an isosceles triangle?
Ans: If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an Isosceles triangle, its perimeter is given by \(=a+a+b=(2a+b)\) units.
If the perimeter is known and congruent sides are known, we can find the other side solving for \(b\) using the above expression.

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