What Is an Isosceles Right Angled Triangle?
As learned in lower grades, an isosceles triangle is a type of triangle that has two sides of equal length and a right triangle is a triangle that has one right angle (90 degrees). So, an isosceles right triangle is one that has two congruent sides and a right angle. The two angles opposite the congruent sides of an isosceles triangle are equal in measure. So, for an isosceles right triangle one angle is 90 degrees and the other two angles are congruent.
Let us assume the other two angles are x° each, then as per angle sum property of a triangle:
90° + x°+ x° = 180°
Solving for x, x° = 45°
Therefore, the angle measures of an isosceles right triangle are 90°, 45° and 45° respectively.
[Note: The Pythagorean theorem is associated with a right triangle. So, the same is also applicable to an isosceles right triangle.]
Formula Associated with 45-45-90 Triangle
The triangle ABC shown in the image is an isosceles right triangle with 90° at vertex A, 45° at vertex B and 45° at vertex C.
Here, side BC is the hypotenuse and congruent sides, AC and AB are the legs of the triangle. A 45-45-90 triangle exhibits a special relationship among the three side measures. With angle measures given, we can say that the three angles are in the ratio 1:1:2 and the two sides opposite 45 degree angles are both equal in length.
45-45-90 Triangle Rule
The 45-45-90 triangle rule states that the three sides of the triangle are in the ratio 1:1:\(\sqrt{2}\). So, if the measure of the two congruent sides of such a triangle is x each, then the three sides will be x, x and \(\sqrt{2}x\).
This rule can be proved by applying the Pythagorean theorem.
For the triangle ABC,
Hypotenuse, BC = \(\sqrt{2}x\)
Leg, AC = x and
Leg, AB = x
\(AB^{2} + AC^{2} = BC^2{}\)
\(\Rightarrow x^{2} + x^{2} = \left( \sqrt{2} x\right)^{2}\)
\(\Rightarrow 2x^{2} = 2x^{2}\)
LHS = RHS
So, AB:AC:BC \(= x : x :\sqrt{2}x\text{ }or\text{ }1 : 1 : \sqrt{2}.\)
This rule can be used to find the unknown side measures of an isosceles right triangle.
Solved Examples
Example 1: Find the missing side measures in the following triangles:
1. Triangle PQR, ∠Q = 45°, Hypotenuse = 15 centimeters
2. Triangle MNO, ∠O = 45°, NO = 6 millimeters
Solution:
- Triangle PQR, ∠Q = 45°, Hypotenuse = 15 centimeters
The sides of the triangle are in the ratio: PQ : PR : QR = x : x : \(\sqrt{2}x\)
So, PQ : \(PQ : PR : QR = x : x : \sqrt{2}x = x : x : 15 \) [Substitute]
\(\Rightarrow \sqrt{2}x = 15\)
\(\Rightarrow x = \frac{15}{\sqrt{2}}\)
Hence, the other two sides of triangle PQR are: \(\frac{15}{\sqrt{2}}\) centimeters each.
2. Triangle MNO, ∠O = 45°, NO = 6 millimeters
The sides of the triangle are in the ratio: MN : NO : OM = x : x : \(\sqrt{2}x\)
So, MN : NO : OM = x : x : \(\sqrt{2}x\) = x : 6 :\(\sqrt{2}x\)
\(\Rightarrow x = 6\text{ }mm\)
\(\Rightarrow \sqrt{2}x = 8.484\text{ }mm\)
Hence, the other two sides of triangle MNO are: 6 millimeters and 8.484 millimeters.
Example 2: Julie is making a triangular poster for an art class project as shown in the image. She needs help with the side measurements of the poster to buy colored tape of enough length for the border. She knows that the longest side of the poster is 8 inches in length. Help Julie find the measures of the other two sides of the poster. Take \(\sqrt{2}\) as 1.414.
Solution:
Let’s consider this triangular poster as a triangle, PQR. The angles are also shown in the image.
The sides of the triangle are in the ratio: \(PQ : PR : QR = x : x :\sqrt{2}x\)
So, \(PQ : PR : QR = x : x :\sqrt{2}x = x : x : 8\) [Substitute]
\(\Rightarrow \sqrt{2}x = 8\)
\(\Rightarrow x = 5.66\text{ }in\)
So, the other two sides of the triangle PQR are: 5.66 inches each
Hence, Julie needs colored tape to border the poster that with sides measuring as 8 inches,5.66 inches and 5.66 inches.
Example 3: Find the missing sides and check your solution with the Pythagorean theorem.
Solution:
The sides of the triangle are in the ratio: AB : BC : CA = x : x : \(\sqrt{2}x\)
So, AB : BC : CA = x : x : \(\sqrt{2}x\) = x : 3 : 2x [Substitute]
\(\Rightarrow x = 3\)
\(\Rightarrow \sqrt{2}x = 4.242\)
So, AB : BC : CA = x : x : \(\sqrt{2}x\)= 3 : 3 : 4.242
Now let’s apply the Pythagorean theorem to check our answer.
\(AB^{2} + BC^{2} = AC^{2}\)
\(\Rightarrow 9 + 9 = 18\)
Hence, this is proof that triangle ABC is a right triangle and the measures of the missing sides are correct.
All three angles of an equilateral triangle are 60 degrees each, so a right triangle cannot be an equilateral triangle.
Triangles with an angle measure greater than 90 degrees are called obtuse angled triangles or simply obtuse triangles.
A triangle cannot have two obtuse angles, as otherwise the sum of the angles of the triangle will exceed 180 degrees and the angle sum property of the triangle will not hold true.
The Pythagorean theorem states that the sum of the squares of the lengths of the two adjacent sides of a right triangle is equal to the square of the length of the hypotenuse.