Video Transcript
Find the length of line segment
π·π΅ rounded to the nearest hundredth, if needed.
In this problem, we need to find
one of the lengths which makes up part of the triangle. We can observe that in the figure
we have two triangles. It may be useful to consider if
these triangles are in fact similar triangles as this would help us establish the
unknown length. We can recall that similar
triangles have corresponding angles congruent and corresponding sides in
proportion.
Now, sometimes, we might find it
hard to visualize the two separate triangles in a figure like this. So, it can be helpful to draw
another sketch of the two triangles separately. For example, here, we can see the
larger triangle π΄πΆπ΅ and the smaller triangle π΄π·πΈ drawn in the same orientation
so that the angles at πΆ and π· are in the same position.
We can write on the lengthsβ
information from the original diagram that π΄π· is 10 centimeters, π΄πΈ is 13
centimeters, and π΄πΆ is the sum of the lengths of 13 and 12 centimeters, which is
25 centimeters. We can note from the markings on
the diagram that there are two congruent angles: angle πΆ and angle π·. And we can observe that the angle
at π΄ is a common or shared angle. So, its measure will be equal in
the two triangles.
So, letβs return to considering if
the triangles are similar. Recall that one way we can prove
triangles are similar is by demonstrating that all pairs of corresponding angles are
congruent. In our diagram, we know that two
pairs of angles are congruent. But in fact, knowing that two pairs
of angles are congruent is enough to show that all three pairs of angles are
congruent because we know that the angle measures in a triangle sum to 180
degrees. And since we know that the other
two pairs of corresponding angles are congruent, then the remaining pair of angles
in each triangle must be congruent.
So, all corresponding angles are
congruent. And importantly, we can now say
that these two triangles are similar. We can write this similarity
relationship as triangle π΄πΆπ΅ is similar to triangle π΄π·πΈ.
Now remember, we want to find the
length of the line segment π·π΅. π·π΅ is part of the longer line
segment π΄π΅. And we know that the other part of
this line segment, π΄π·, has a length of 10 centimeters. We can note that if we knew the
length of the line segment π΄π΅, this would help us work out the length of line
segment π·π΅. Because the triangles are similar,
then line segment π΄πΈ in triangle π΄π·πΈ is corresponding to line segment π΄π΅. And in similar triangles,
corresponding sides are in proportion. So, we need to find a pair of
corresponding sides whose lengths we are given. That would be line segments π΄πΆ
and π΄π·. We can write that π΄πΆ over π΄π· is
equal to π΄π΅ over π΄πΈ.
We could alternatively have written
these fractions or proportions with the numerators and denominators flipped. The important bit about writing the
proportion is that we keep the sides on each triangle either as both on the
numerators or both on the denominators. We can then fill in the lengths
that we know. This gives us 25 over 10 equals
π΄π΅ over 13. We might simplify the left side
first to five over two before multiplying both sides by 13, which gives us that π΄π΅
equals 65 over two or 32.5 centimeters.
But we havenβt finished the
question yet. Weβve worked out the length of the
longer line segment π΄π΅, but we really want to know the length of line segment
π·π΅. So π·π΅ is 32.5 centimeters
subtract 10 centimeters, which is 22.5 centimeters.
Therefore, by first proving that
the triangles are similar, we have determined that the length of line segment π·π΅
is 22.5 centimeters. And as this value has one decimal
place, we donβt need to round the answer to the nearest hundredth.