About Isosceles Right Angle Triangle
Right Angle Triangle
A right angle triangle is a triangle whose one angle is 90 degrees. As we know that the 90 degree is also called a right angle, therefore, a triangle with one angle 90 degree is called a right angle triangle.
Isosceles Right Angle Triangle
An isosceles right angle triangle is a right angle triangle that has equal leg lengths. Since the legs of a right angle triangle are called the perpendicular and base, therefore, we can say that perpendicular and base are congruent in an isosceles right angle triangle.
Area of Isosceles Triangle
As we know that the area of triangle formula is \( \frac{1}{2} \times b \times h \) square units
Where b is the base of the triangle and h is the altitude of the triangle.
In an isosceles triangle the base and perpendicular length are congruent. Suppose the length of each leg is a unit, so we can modify the area formula.
Area of triangle \( ABC = \frac{1}{2} \times a \times a \)
\( = \frac{1}{2} a^2 \)
So, the area of an isosceles right angle triangle is \( \frac{1}{2} a^2 \) square unit. Where a is the length of equal legs.
Perimeter of Isosceles Right Angle Triangle
Perimeter of any figure is the total length of its boundary. Since an isosceles right angle triangle has a hypotenuse and equal legs, so we add them to get the perimeter. The units of the perimeter of an isosceles right angle triangle are inches(in), yards(yd), and meters(m).
Suppose the length of the hypotenuse is h and the length of the legs is a.
Perimeter of an isosceles right angle triangle ABC, \( P = AB + BC + AC \)
\( = a + a + h \)
\( = 2a + h \)
So, the perimeter of an isosceles right angle triangle is \( 2a + h \).
Where ‘a’ is the length of the equal sides and ‘h’ is the length of the hypotenuse.
In addition the length of the hypotenuse (h) can be found by the formula
Hypotenuse \( (h) = \sqrt{(a^2 + a^2)} \)
\( = \sqrt{2a^2} \)
\( = \sqrt{2}a \)
Where a is the equal sides of an isosceles right angle triangle.
Rapid Recall
In isosceles right angle triangle
Area, \( A = \frac{1}{2}a^2 \)
Perimeter, \( P = 2a + h \)
Solved Examples
Example 1: Find the area of the given triangle.
Solution :
\( A = \frac{1}{2}a^2 \) [Area formula]
\( A = \frac{1}{2} \times 6^2 \) [Substitute \( 6 \) for a]
\( A = 18 \) [Solve]
So, the area of a given triangle is \( 18 \) square inches.
Example 2: Annie cut some cardboard into the shape of a right triangular shape as shown in the image. Find the amount of color paper required for the project.
Solution:
\( A = \frac{1}{2}a^2 \) [Area formula]
\( A = \frac{1}{2} \times 20^2 \) [Substitute \( 20 \) for a]
\( A = 200 \) [Solve]
So, \( 200 \) square centimeter color paper is required for the project.
Example 3: Find the perimeter of a given triangle.
Solution:
\( P = 2a + h \) [Formula for perimeter]
\( P = 2\times 12 + 16.97 \) [Substitute \( 12 \) for a and \( 16.97 \) for h]
\( P = 24 + 16.97 \) [Multiply \( 2 \) and \( 12 \)]
\( P = 40.97 \)
So, the perimeter of a given triangle is \( 40.97 \) millimeters.
A right angle triangle with equal legs is called an isosceles right angle triangle.
There are two equal sides in the isosceles right angle triangle.
The sum of interior angles of an all triangle is 180 degrees.
The unequal side of the isosceles right angle triangle is called the hypotenuse.
The perimeter of an isosceles right triangle is, p = 2a + h where a is the equal side length and h is the length of the hypotenuse.