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First note that every isosceles right triangle is similar. They all have the same shape, they're just scaled differently. Now consider the isosceles right triangle with the two equal sides being $1$ unit. These have to be the mutually perpendicular sides (technically called the catheti) because the hypotenuse is strictly longer than any other side. Apply Pythagoras' Theorem. The hypotenuse $h$ is given by $h^2 = 1^2 + 1^2$, giving $h = \sqrt 2$. You now know the ratios of the sides of every isosceles right triangle, i.e. $1:1:\sqrt 2$. To apply it, if you're given that the hypotenuse is $13$, then the other two sides are each $\frac{13}{\sqrt 2} $. This is often rearranged to $\frac{13\sqrt 2}{2} $ to avoid an irrational number in the denominator.
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. 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 HypotenuseThe 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 PropertiesIsosceles right triangle follows almost similar properties to an isosceles triangle. Let's look into the list of properties followed by the isosceles right triangle.
Isosceles Right Triangle FormulaIsosceles 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 = 2x2 Thus, l = x√2 units Perimeter of Isosceles Right Triangle FormulaThe 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 AreaThe 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. Related Articles Check these articles related to the concept of an isosceles right triangle in geometry.
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FAQs on Isosceles Right TriangleAn 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. |
What is the length of the legs of an isosceles right triangle if the hypotenuse is 8 units?
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