A triangle has vertices of the
points 𝐴 four, one; 𝐵 six, two; and 𝐶 two, five. Work out the lengths of the sides
of the triangle. Give your answers as surds in their
simplest form. And secondly, is this triangle a
Let’s begin by sketching this
triangle on a coordinate grid. We absolutely don’t need to plot
this triangle accurately. We aren’t going to be measuring the
lengths of any of the lines. We just want to sketch it using the
approximate position of these three points relative to one another.
So the triangle looks a little
something like this. Now, from our sketch, it looks
possible that this could be a right triangle with the right angle at 𝐴. But we can’t confirm this from our
sketch. Let’s consider the first part of
the question. We need to find the lengths of the
three sides of the triangle. And we’ll begin by finding the
length of the side 𝐴𝐵.
We can sketch in a right triangle
below this line using 𝐴𝐵 as its hypotenuse. We can also work out the lengths of
the other two sides in this triangle. The horizontal side will be the
difference between the 𝑥-values at its endpoints. That’s the difference between six
and four, which is two. And the vertical side will be the
difference between the 𝑦-values at its endpoints. That’s the difference between two
and one, which is one.
As we now have the lengths of two
sides in a right triangle and we wish to calculate the length of the third side, we
can apply the Pythagorean theorem, which tells us that, in a right triangle, the sum
of the squares of the two shorter sides is equal to the square of the
hypotenuse. Remember, 𝐴𝐵 is the
hypotenuse. So we have that 𝐴𝐵 squared is
equal to one squared plus two squared. One squared is one and two squared
is four. So adding these values together, we
have that 𝐴𝐵 squared is equal to five.
To find the length of 𝐴𝐵, we need
to square root each side of this equation. And remember at this point, we’ve
been told to give our answer as a surd. So we have that 𝐴𝐵 is equal to
root five. We can find the lengths of the
other two sides of the triangle in the same way. We sketch in a right triangle below
the line 𝐵𝐶. And we see that it has a horizontal
side of four units and a vertical side of three units.
𝐵𝐶 is the hypotenuse of this
triangle. So applying the Pythagorean
theorem, we have that 𝐵𝐶 squared is equal to three squared plus four squared. That’s nine plus 16, which is equal
to 25. 𝐵𝐶 is therefore equal to the
square root of 25, which is simply the integer five. In the same way, 𝐴𝐶 is the
hypotenuse of a right triangle with shorter sides of two and four units. So 𝐴𝐶 is equal to the square root
of 20, which simplifies to two root five.
So we’ve answered the first part of
the question. And now we need to determine
whether this triangle is a right triangle. Well, if it is, then the
Pythagorean theorem will hold for its three side lengths. Now we suspect it that the right
angle was at 𝐴, which would make 𝐵𝐶 the hypotenuse of the triangle if it is
indeed a right triangle.
We therefore want to know whether
𝐵𝐶 squared is equal to 𝐴𝐵 squared plus 𝐴𝐶 squared. Well, we can in fact use the
squared side lengths. We know that 𝐵𝐶 squared is
25. We know that 𝐴𝐵 squared is
five. And we know that 𝐴𝐶 squared is
20. So is it true that 25 is equal to
five plus 20? Yes, of course, it’s true, which
means that the Pythagorean theorem holds for this triangle. And therefore, it is indeed a right
triangle. So we’ve completed the problem. We have the three side lengths. 𝐴𝐵 equals root five, 𝐵𝐶 equals
five, and 𝐴𝐶 equals two root five. And we’ve determined that the
triangle is a right triangle.