### Video Transcript

Triangles ๐ด๐ต๐ถ and ๐ด prime ๐ต
prime ๐ถ prime are similar. Work out the measure of angle
๐ฅ. Work out the value of ๐ฆ. Work out the value of ๐ง.

In this question, weโre 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 ๐ด
prime denoted by the ๐ฅ, then we need to work out which angle in triangle ๐ด๐ต๐ถ is
corresponding to this one. Sometimes in diagrams, this isnโt
always clear, but we can use the order of the letters to help us.

The angle at ๐ด prime will
correspond with the angle at ๐ด. We can write this more formally as
the angle ๐ถ prime ๐ด prime ๐ต prime is corresponding to the angle at ๐ถ๐ด๐ต, this
one in pink. Both of these angles are equal, and
theyโre 74.5 degrees. So our answer for the first part of
this question is that angle ๐ฅ 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, weโre asked to find the length of ๐ฆ. Weโll 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 ๐ด prime
๐ต prime is five and the length ๐ด๐ต is two. To work out the scale factor from
the smaller triangle to the larger triangle, weโll 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 ๐ฆ, which we want to
calculate on the larger triangle on the line ๐ต prime ๐ถ prime, corresponds with the
line ๐ต๐ถ of length three on the smaller triangle. So, we can calculate ๐ฆ 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 ๐ฆ equals 7.5.

There is an alternative method we
couldโve used to work out the value of ๐ฆ. As we know that our triangles are
similar, our lengths will all be in the same proportion. Looking at the lengths ๐ด prime ๐ต
prime, which is five, and ๐ด๐ต, which is two, we can say that five over two is equal
to ๐ฆ over three. As these triangles are similar, we
know that itโs the same proportion between the lengths five and two as it would be
between ๐ฆ and three. We can then take the cross product,
and so two times ๐ฆ is two ๐ฆ equals five times three, which is 15. And if two ๐ฆ is 15, then ๐ฆ is
half of that. So, ๐ฆ is 7.5, confirming the
answer that we found by working out the scale factor.

Letโs take a look at the final part
of this question to find the value of ๐ง. 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, weโd 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 weโre
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
๐ง on the line ๐ด๐ถ, we take the corresponding length ๐ด prime ๐ถ 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 ๐ง is 1.6.

We couldโve worked out this final
question by using the original scale factor. We wouldโve set up an equation that
said ๐ง times five over two equals four. We wouldโve then rearranged this to
find the value of ๐ง. This method does track very closely
to finding the reverse scale factor however. Both methods would confirm that ๐ง
is 1.6.