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This page looks at how we can apply the above formula, given WE HAVE A RIGHT ANGLED TRIANGLE.
Given we have the lengths of 2 out of the three sides of a triangle AND one of the angles in the triangle is 90 degrees, in other words a right angle, we can find the length of the third side.
If we have this criteria, we are able to use this formula.
A right angled triangle, like any triangle, has three sides, each have their own lengths which we can represent with the letters a, b and c. The longest side (in length) is called the hypotenuse, we represent the length of the longest side with c. This will always be the diagonal side.
It is important that you understand that the formula shows us how to work out the length of the hypotenuse (the longest side). The hypotenuse will always be the diagonal side of a right angled triangle.
We can represent the lengths of the other two sides with a and b. It does not matter which side's length we represent with a and b, as long as the longest side's length, in the right angled triangle, is called c.
Whether we use a, b or c as letters to represent the lengths or we use the letters f, m and n, it does not matter. The main point is that the formula tells us is that given we have the lengths of two side of a right angled triangle, we can work out the length of the third side using the formula.
Thus, the formula tells us to work out the length of the hypotenuse, we can square the lengths of the sides, which aren't the hypotenuse and sum the squares together. The square root of the sum is then found, to find the length of the hypotenuse. Knowing this, if there is a situation where we know the length of the hypotenuse and know the length of another side, we can find length of the third side.
This will be clearer when you understand the questions below.
Given we have the lengths of 2 out of the three sides of a triangle AND one of the angles in the triangle is 90 degrees, in other words a right angle, we can find the length of the third side.
If we have this criteria, we are able to use this formula.
A right angled triangle, like any triangle, has three sides, each have their own lengths which we can represent with the letters a, b and c. The longest side (in length) is called the hypotenuse, we represent the length of the longest side with c. This will always be the diagonal side.
It is important that you understand that the formula shows us how to work out the length of the hypotenuse (the longest side). The hypotenuse will always be the diagonal side of a right angled triangle.
We can represent the lengths of the other two sides with a and b. It does not matter which side's length we represent with a and b, as long as the longest side's length, in the right angled triangle, is called c.
Whether we use a, b or c as letters to represent the lengths or we use the letters f, m and n, it does not matter. The main point is that the formula tells us is that given we have the lengths of two side of a right angled triangle, we can work out the length of the third side using the formula.
Thus, the formula tells us to work out the length of the hypotenuse, we can square the lengths of the sides, which aren't the hypotenuse and sum the squares together. The square root of the sum is then found, to find the length of the hypotenuse. Knowing this, if there is a situation where we know the length of the hypotenuse and know the length of another side, we can find length of the third side.
This will be clearer when you understand the questions below.
Figure 1 for Question 1
Figure 1 for Question 1
Given we have a right angled triangle, we can label the longest side c, and find the lengthof the side using a2 + b2 = c2.
Question 1: A right angled triangle has two sides with length 4cm and 3cm as shown in figure 1. What is the length of the side which we do not know?
From the diagram, we can see that the third side is the longest side (hypotenuse) as thisis the diagonal side of the right angled triangle. So we can apply the theorem to find thelength of the hypotenuse.
If we let c = length of hypotenuse, a = 3 and b = 4 we can substitute the values intothe formula.
32 + 42 = 9 + 16 = 25
so
c2= 25
(Square root both sides)
√(c2) = ±√(25)
c = ±5
The square root of 25 is 5 and -5. So we have two values for c.
BUT WAIT!
Even though we have two values for c, one of the values is negative. This cannot be the answer as length cannot be negative. So in this case, c is equal to 5.
Thus the length of the hypotenuse is 5cm.
Question 1: A right angled triangle has two sides with length 4cm and 3cm as shown in figure 1. What is the length of the side which we do not know?
From the diagram, we can see that the third side is the longest side (hypotenuse) as thisis the diagonal side of the right angled triangle. So we can apply the theorem to find thelength of the hypotenuse.
If we let c = length of hypotenuse, a = 3 and b = 4 we can substitute the values intothe formula.
32 + 42 = 9 + 16 = 25
so
c2= 25
(Square root both sides)
√(c2) = ±√(25)
c = ±5
The square root of 25 is 5 and -5. So we have two values for c.
BUT WAIT!
Even though we have two values for c, one of the values is negative. This cannot be the answer as length cannot be negative. So in this case, c is equal to 5.
Thus the length of the hypotenuse is 5cm.
Figure 2 for Question 2
Figure 2 for Question 2
Question 2: Find the missing length of the right angled triangle in figure 2.
Lets represent the lengths of the sides first, so we can substitute the letters in a2 + b2 = c2 with the values of the length. We can see that the hypothenuse is given so we know c = 7.
Once we have labelled the hypotenuse, we can represent the other two sides with a and b. Lets choose to labelthe side with length 5cm with a and the length of the side with the missing length b.
So we have
c = 7, a = 5 and b = The length we need to find
If we substitute the c with 7 and the a with 5 in
a2 + b2 = c2
We get
52 + b2 = 72
so now lets solve to find b.
52 + b2 = 72
(Subtract 52 from both sides)
52 +b2 - 52= 72 - 52
b2 = 72 - 52
b2 = 49 - 25 = 24
(Square root both sides)
√(b2) = ±√(24)
b = ±√(24)
The square root of 24 is √(24) and -√(24), so we have two values for b.
BUT WAIT!
Even though we have two values for b, one of the values is negative. This cannot be the answer as length cannot be negative. So in this case, b is equal to √(24) (in surd form).
b = √(24) = 4.9cm (to 1 decimal place)
Lets represent the lengths of the sides first, so we can substitute the letters in a2 + b2 = c2 with the values of the length. We can see that the hypothenuse is given so we know c = 7.
Once we have labelled the hypotenuse, we can represent the other two sides with a and b. Lets choose to labelthe side with length 5cm with a and the length of the side with the missing length b.
So we have
c = 7, a = 5 and b = The length we need to find
If we substitute the c with 7 and the a with 5 in
a2 + b2 = c2
We get
52 + b2 = 72
so now lets solve to find b.
52 + b2 = 72
(Subtract 52 from both sides)
52 +b2 - 52= 72 - 52
b2 = 72 - 52
b2 = 49 - 25 = 24
(Square root both sides)
√(b2) = ±√(24)
b = ±√(24)
The square root of 24 is √(24) and -√(24), so we have two values for b.
BUT WAIT!
Even though we have two values for b, one of the values is negative. This cannot be the answer as length cannot be negative. So in this case, b is equal to √(24) (in surd form).
b = √(24) = 4.9cm (to 1 decimal place)
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