Triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵
prime 𝐶 prime are similar. Work out the measure of angle
𝑥. Work out the value of 𝑦. Work out the value of 𝑧.
In this question, we’re told that
our two triangles are similar, which means that corresponding angles are congruent
and corresponding sides are in proportion. If we look at this angle at 𝐴
prime denoted by the 𝑥, then we need to work out which angle in triangle 𝐴𝐵𝐶 is
corresponding to this one. Sometimes in diagrams, this isn’t
always clear, but we can use the order of the letters to help us.
The angle at 𝐴 prime will
correspond with the angle at 𝐴. We can write this more formally as
the angle 𝐶 prime 𝐴 prime 𝐵 prime is corresponding to the angle at 𝐶𝐴𝐵, this
one in pink. Both of these angles are equal, and
they’re 74.5 degrees. So our answer for the first part of
this question is that angle 𝑥 is 74.5 degrees. It can be tempting to think that
because the triangle is larger, that the angle must also be larger. But remember that the sum of the
angles in a triangle is always 180 degrees.
In the second part of this
question, we’re asked to find the length of 𝑦. We’ll need to work out the
proportion of the lengths or the scale factor that takes us from the smaller
triangle to the larger triangle. We can use a given pair of
corresponding sides. Here, we have the length 𝐴 prime
𝐵 prime is five and the length 𝐴𝐵 is two. To work out the scale factor from
the smaller triangle to the larger triangle, we’ll take our new length of five and
divide it by the original length of two. Therefore, if we want to find a
length on the longer triangle, we take the corresponding length on the smaller
triangle and multiply it by five over two.
So, the length 𝑦, which we want to
calculate on the larger triangle on the line 𝐵 prime 𝐶 prime, corresponds with the
line 𝐵𝐶 of length three on the smaller triangle. So, we can calculate 𝑦 by
multiplying the length three by the scale factor of five over two. Three times five is 15, and 15 over
two simplifies to 7.5. And so, our answer to the second
part of this question is 𝑦 equals 7.5.
There is an alternative method we
could’ve used to work out the value of 𝑦. As we know that our triangles are
similar, our lengths will all be in the same proportion. Looking at the lengths 𝐴 prime 𝐵
prime, which is five, and 𝐴𝐵, which is two, we can say that five over two is equal
to 𝑦 over three. As these triangles are similar, we
know that it’s the same proportion between the lengths five and two as it would be
between 𝑦 and three. We can then take the cross product,
and so two times 𝑦 is two 𝑦 equals five times three, which is 15. And if two 𝑦 is 15, then 𝑦 is
half of that. So, 𝑦 is 7.5, confirming the
answer that we found by working out the scale factor.
Let’s take a look at the final part
of this question to find the value of 𝑧. We know that we go from the smaller
triangle to the larger triangle by multiplying the lengths by five over two. But what happens in the reverse
direction? In this case, we’d have to perform
the inverse operation. We could informally say that we
need to divide the lengths by five over two, but scale factors should be given as a
multiplier. We can recall that when we’re
dividing by a fraction, this is equivalent to multiplying by the fraction
flipped. So, our scale factor from the large
triangle to the small triangle would be two-fifths.
In order to work out the length of
𝑧 on the line 𝐴𝐶, we take the corresponding length 𝐴 prime 𝐶 prime, which is
four, and multiply it by the scale factor of two-fifths. Four times two is eight, and
eight-fifths is equivalent to 1.6. So, our answer for the final part
of this question is that 𝑧 is 1.6.
We could’ve worked out this final
question by using the original scale factor. We would’ve set up an equation that
said 𝑧 times five over two equals four. We would’ve then rearranged this to
find the value of 𝑧. This method does track very closely
to finding the reverse scale factor however. Both methods would confirm that 𝑧