Determine the lengths of 𝐴𝑁 and 𝐵𝑁.
Within this diagram, we can see that we have two triangles: triangles 𝐴𝐵𝐶 and 𝐴𝑁𝑃, each with a mixture of sides and angles marked. We can see that the angles in the two triangles are the same; they both have an angle of 104 degrees and an angle of 39 degrees, which means the third angle must also be the same as the angles sum in a triangle is always 180 degrees.
We now consider whether these two triangles are congruent to each other. As if they are, this will help us with calculating the length of 𝐴𝑁 and 𝐵𝑁. Let’s start by listing what we know. The two angles of 104 degrees are the same, so angle 𝐵𝐶𝐴 is equal to angle 𝑁𝐴𝑃. The A in brackets is used to indicate a statement about equal angles. Next, let’s look at the two sides that have been marked as 37 centimetres: sides 𝐶𝐴 and 𝐴𝑃. So we have the statement 𝐶𝐴 is equal to 𝐴𝑃 and the S in brackets indicates that this is a statement about a side.
Now let’s look at the angles of 39 degrees: angles 𝐶𝐴𝑃 and 𝐴𝑃𝑁. We have the angle 𝐶𝐴𝑃 is equal to angle 𝐴𝑃𝑁. And again the A in brackets indicates that this is a statement about angles. So looking at our three statements, we can see that we do have enough information to conclude that these two triangles are congruent to each other. The ASA tells us that this is the angle-side-angle congruence criteria that we can use. So we conclude that triangle 𝐵𝐶𝐴 is congruent to triangle 𝑁𝐴𝑃.
Now how does this help with us answering the question, which was to determine the length of 𝐴𝑁 and 𝐵𝑁. Well, let’s look which pairs of sides correspond with each other between the two triangles. Side 𝐴𝑁 corresponds with side 𝐶𝐵; therefore, 𝐴𝑁 must be equal to 38 centimetres. The third side of each of the triangles must correspond with each other, so that side 𝐴𝐵 and 𝑃𝑁.
We know that 𝑃𝑁 is equal to 59 centimetres, and therefore 𝐴𝐵 is also equal to 59 centimetres. However, the question didn’t ask us to find 𝐴𝐵; it asked us to find 𝐴𝑁, which we’ve done and 𝐵𝑁. In order to calculate 𝐵𝑁, we can take the whole length of 𝐴𝐵 and then subtract the length of 𝐴𝑁, both of which we know. So 𝐵𝑁 is equal to 59 minus 38, which is 21. So we have our answer to the problem. 𝐴𝑁 is equal to 38 centimetres; 𝐵𝑁 is equal to 21 centimetres.