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Isosceles Right Triangle

An isosceles triangle is defined as a triangle that has two sides of equal measure. An isosceles triangle with a right angle is known as an isosceles right triangle. We will be studying the properties and formulas of the isosceles right triangle along with examples in this article.

 1 What is Isosceles Right Triangle? 2 Isosceles Right Triangle Properties 3 Isosceles Right Triangle Formula 4 FAQs on Isosceles Right Triangle

## What is Isosceles Right Triangle?

An isosceles right triangle is defined as a right-angled triangle with an equal base and height which are also known as the legs of the triangle. It is a special isosceles triangle with one angle being a right angle and the other two angles are congruent as the angles are opposite to the equal sides. It is also known as a right-angled isosceles triangle or a right isosceles triangle. The area of an isosceles right triangle follows the general formula of the area of a triangle where the base and height are the two equal sides of the triangle. Let’s look into the image of an isosceles right triangle shown below. If the congruent sides measure x units each, then the hypotenuse or the unequal side of the triangle will measure x√2 units.

### Isosceles Right Triangle Hypotenuse

The hypotenuse of a right isosceles triangle is the side opposite to the 90-degree angle. It is √2 times the length of the equal side of the triangle. So, if the measurement of each of the equal sides is x units, then the length of the hypotenuse of the isosceles right triangle is x√2 units. It is derived using the Pythagoras theorem which you will learn in the section below.

## Isosceles Right Triangle Properties

Isosceles right triangle follows almost similar properties to an isosceles triangle. Let’s look into the list of properties followed by the isosceles right triangle.

• It has one angle measuring 90º.
• The legs of this triangle are perpendicular to each other which are also known as the base and the height.
• The other two angles of an isosceles right triangle are acute and congruent to each other measuring 45° each.
• The sum of all the interior angles is equal to 180°.
• The altitude drawn from the right angle is the perpendicular bisector of the hypotenuse (opposite side).
• The area of an isosceles right triangle is given as (1/2) × Base × Height square units.

## Isosceles Right Triangle Formula

Isosceles right triangle follows the Pythagoras theorem to give the relationship between the hypotenuse and the equal sides. Let’s look into the diagram below to understand the isosceles right triangle formula.

In ∆PQR,
PQ = QR = x units [Equal Sides]
PR = l units [Hypotenuse]
Using Pythagoras theorem,
Hypotenuse2 = Side2 + Side2
PR2 = PQ2 + QR2
l2 = x2 + x2
l2 = 2×2
Thus, l = x√2 units

### Perimeter of Isosceles Right Triangle Formula

The perimeter of an isosceles right triangle is defined as the sum of all three sides. In ∆PQR shown above with side lengths PQ = QR = x units and PR = l units, perimeter of isosceles right triangle formula is given by PQ + QR + PR = x + x + l = (2x + l) units.

Thus, the perimeter of the isosceles right triangle formula is 2x + l, where x represents the congruent side length and l represents the hypotenuse length.

### Isosceles Right Triangle Area

The area of isosceles right triangle follows the general formula of area of a triangle that is (1/2) × Base × Height. In ∆PQR shown above with side lengths PQ = QR = x where PQ represents the height and QR represents the base, the area of isosceles right triangle formula is given by 1/2 × PQ × QR = x2/2 square units.

Thus, the area of the isosceles right triangle formula is x2/2, where x represents the congruent side length.

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## Isosceles Right Triangle Examples

1. Example 1: The equal sides of a right isosceles triangle measures 8 units each. Find the area.

Solution: For an isosceles right triangle, the area formula is given by x2/2 where x is the length of the congruent sides.

Here, x = 8 units

Thus, Area = 82/2 = 32 square units

Therefore, the required area is 32 square units.

2. Example 2: The perimeter of an isosceles right triangle is 10 + 5√2. If the non-congruent side measures 5√2 units then, find the measure of the congruent sides.

Solution: For a right isosceles triangle, the perimeter formula is given by 2x + l where x is the congruent side length and l is the length of the hypotenuse.

Here, l = 5√2 units, Perimeter = 10 + 5√2 units

By using the formula,

2x + l = 10 + 5√2

2x = 10 [Since, l = 5√2]

Thus, x = 5 units

Hence, the length of each congruent side is 5 units.

3. Example 3: If the isosceles right triangle area is 200 square inches, find the length of the equal sides.

Solution: The right isosceles triangle area formula is x2/2 square units, where x is the length of each equal side. Substituting the given value of area, we get,

200 = x2/2

400 = x2

x = 20

Therefore, the length of each congruent side is 20 inches.

## FAQs on Isosceles Right Triangle

### What is an Isosceles Right Triangle?

An isosceles right triangle is defined as a triangle with two equal sides known as the legs, a right angle, and two acute angles which are congruent to each other.

### Can Isosceles Triangles be Right?

Yes, any isosceles triangle that has one angle measuring 90° and the other two angles congruent to each other (measures 45° each) can be an isosceles right triangle.

### How to Find the Hypotenuse of an Isosceles Right Triangle?

The hypotenuse of an isosceles right triangle is the longest side of the triangle which lies opposite to the right angle. If the measure of each of the equal sides is ‘a’ units, then the length of the hypotenuse is a√2 units.

### How to Find the Area of Isosceles Right Triangle?

The area of an isosceles right triangle is found using the formula side2/2 where the side represents the congruent side length. For example, the area of an isosceles right triangle having the side length of 4 units each is equal to 42/2 = 8 square units.

### What are the Angles of an Isosceles Right Triangle?

The angles of an isosceles right triangle are equal to 90°, 45°, and 45°.

### What is the Smallest Angle in an Isosceles Right Triangle?

The measure of the three angles of an isosceles right triangle are 90°, 45°, and 45°. So, the smallest angle is 45°.

### How to Find Perimeter of Isosceles Right Triangle?

The perimeter of an isosceles right triangle is found by adding all three sides of the triangle. For example, the perimeter of an isosceles right triangle having the base and height measuring ‘a’ units and the hypotenuse measuring ‘b’ units is equal to a + a + b or (2a + b) units.

How many Lines of Symmetry does an Isosceles Right Triangle have?

An isosceles right triangle has one line of symmetry that bisects the right angle and is the perpendicular bisector of the hypotenuse.

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