What is a 45-45-90 triangle?
As you would have previously learnt, there are three types of triangles: equilateral, isosceles, and scalene. The 45-45-90 is an isosceles right triangle since it has two opposite sides of equal length and the angle between the two equal sides is 90°, but this triangle is special because it has some other unique properties. We call this the 45-45-90 triangle, because the triangle has the angles 45°, 45°, and 90° like so:
How to use this concept?
We can use this concept to simplify how to find certain quantities or lengths. For example, to begin we know that the ratio of the sides are 1:1:√2 and the ratio of the angles are 1:1:2. In addition, one of the key properties is that the hypotenuse (longest side of the triangle) is the length of the of either side times the square root of 2 (i.e. hyp = side x√2). Using this formula, if you know either one of the quantities, you can find the other quantity. For example, if I knew the length of the hypotenuse, we would divide hypotenuse by square root of two to find the length of the (equal) sides.
How to use this concept?
We can use this concept to simplify how to find certain quantities or lengths. For example, to begin we know that the ratio of the sides are 1:1:√2 and the ratio of the angles are 1:1:2. In addition, one of the key properties is that the hypotenuse (longest side of the triangle) is the length of the of either side times the square root of 2 (i.e. hyp = side x √2). Using this formula, if you know either one of the quantities, you can find the other quantity. For example, if I knew the length of the hypotenuse, we would divide hypotenuse by square root of two to find the length of the (equal) sides.
Sample Math Problems
Problem
Identify which of these are a 45-45-90 triangle:
A)
B)
C)
D)
Solution
Given that one of the angles is a 90° right angle and that the sum of all angles in a triangle is 180°, we know that the other two angles must add up to 90° (180° - 90° = 90°). And because we can see there are two sides of equal length, we know this is an isosceles triangle, therefore, we know that the angles opposite to the equal sides are also the same. So, we know the last two (equal) angles must add to 90°:
90° = angle #1 + angle #2
90° = 2 • angle #1
(both angles are same, i.e. angle #1 = angle #2)
angle # 1 = 45°
Since angle #1 = angle #2, angle #2 = 45°.
Problem
Given a triangle with two sides of the same length of 5 cm, what is the length of the hypotenuse side of the triangle?
Solution
5√2 cm.
Using what we learnt about 45-45-90 triangles, we know that a 45-45-90 triangle is an isosceles right triangle, so the given triangle will have the same properties as a typical 45-45-90 triangle. One of the distinct properties is a formula that connects the length of the leg/equal sides and the length of the hypotenuse by the following formula:
hyp = side x √2
So now that we know the length of a side, we can find the hypotenuse:
hyp = side x √2
hyp = 5 x √2
hyp = 5√2 cm
Problem
Given a right-angled triangle and the length of the hypotenuse is 12 cm, what are the lengths of the two other sides of the triangle?
Solution
From the question we know that the right-angled triangle, a.k.a. a 45-45-90 triangle, has a hypotenuse of length 12 cm. Based on a unique formula that connects the length of the leg/equal sides and the length of the hypotenuse of a 45-45-90 triangle, we can show the following:
Problem
Find the missing sides of the following triangle:
Solution
b = 14 and a = 14√2
From the diagram, we can see that two of the angles are 90° and 45°, which means that the last angle must be:
180° = 90° + 45° + last angle (all angles in a triangle add to 180°)
180° - 90° + 45° = last angle
45° = last angle
So now we know we have a 45-45-90 right angled triangle, which means we can apply what we learnt about these triangles. Firstly, we can say that this triangle is isosceles because the opposite angles are the same, so the sides must be equal. Thus, the length of side b must be 14. Now to find a, we can apply the following formula we learnt:
hyp = side x √2
hyp = 14√2
a = 14√2
