Video Transcript
𝐴𝐵𝐶𝐷 is a rhombus where the
side length is 28 meters and the measure of angle 𝐴𝐵𝐶 is 129 degrees and six
minutes. Find the lengths of the diagonals,
giving the answer to three decimal places.
In this question, we’re given a
rhombus 𝐴𝐵𝐶𝐷 with a side length of 28 meters. And we’re told the measure of one
of its angles. Angle 𝐴𝐵𝐶 has a measure of 129
degrees and six minutes. We need to use this information to
determine the lengths of the diagonals of this rhombus. We need to give our answers to
three decimal places. To answer this question, it’s
always a good idea to sketch the information we’re given. So we need to sketch rhombus
𝐴𝐵𝐶𝐷. And to do this, we recall that a
rhombus is a quadrilateral where all of the sides have the same length.
This gives us a shape like the
following where there’s a couple of things we need to notice. First, all of the sides will have
length 28 meters. Next, we’re given one angle in our
rhombus. Angle 𝐴𝐵𝐶 has a measure of 129
degrees and six minutes. So it’s a good idea in our sketch
to make sure that this angle is obtuse. Finally, we’re told that 𝐴𝐵𝐶𝐷
is the quadrilateral. This means vertex 𝐴 connects to
vertex 𝐵, which connects to 𝐶, which connects to 𝐷, which connects back to
𝐴. This just tells us the order we
should label the vertices in our quadrilateral.
Now that we sketched our rhombus,
let’s ask the question “what does it mean by the diagonals of this
quadrilateral?” The diagonals of a shape are
connections between two vertices which are not sides. In particular for this
quadrilateral, the diagonals will be 𝐴𝐶 and 𝐵𝐷. To help us determine these lengths,
let’s start by filling in some extra information we can determine by using
properties of rhombuses. First, opposite angles in a rhombus
are equal. So the measure of angle 𝐴𝐷𝐶 is
129 degrees and six minutes. Next, we could determine the
measure of our other two angles. They add to 129 degrees and six
minutes to make 180 degrees.
However, we’ll see that this is not
technically necessary. Let’s start by adding in the line
segment 𝐴𝐶. We can now see the triangle 𝐴𝐵𝐶
and triangle 𝐴𝐷𝐶 are isosceles triangles. And remember, in an isosceles
triangle, the angles opposite the equal sides have the same measure. If we call this measure 𝑥, then we
can use the fact that the sum of the internal angles of a triangle will add to 180
degrees to determine the value of 𝑥. We have that two 𝑥 plus 129
degrees and six minutes is equal to 180 degrees. We can solve this equation for
𝑥. We want to solve this equation for
𝑥. We’ll start by subtracting 129
degrees and six minutes from both sides of the equation. This gives us two 𝑥 is equal to
180 degrees minus 129 degrees and six minutes, which we can calculate is 50 degrees
and 54 minutes.
Now we can solve this equation for
𝑥 by dividing through by two, we get 𝑥 is 50 degrees and 54 minutes divided by
two, which we can calculate is 25 degrees and 27 minutes. We can then use this to update our
sketch. And it’s worth noting we would get
the exact same story in our other isosceles triangle. The other two angles would be the
same.
Let’s now add in the other diagonal
into our sketch. We recall diagonals in a rhombus
meet at right angles. This gives us the following. We can now see that we’ve split our
rhombus into four smaller right triangles. And each of these right triangles
shares two angles and a side. In other words, all four of these
triangles are congruent. And in particular, this gives us a
useful result. If we call the point of
intersections between the diagonals 𝐸, then 𝐵𝐸 must have the same length as
𝐸𝐷. And 𝐴𝐸 must also have the same
length as 𝐸𝐶. Therefore, we can determine the
lengths of the diagonals by just finding the length of one of these right
triangles.
So to do this, let’s just sketch
one of the smaller triangles. Let’s choose triangle 𝐵𝐸𝐶. This gives us the following. We now have a right triangle where
we know one of the non-right angles and one of the side lengths. We can use right triangle
trigonometry to determine the other two side lengths.
However, before we do this, we note
that our angle is given in degrees and minutes. And when we’re using trigonometry,
it’s a good idea to rewrite our number in terms of degrees. This is because calculators work in
degrees and not degrees and minutes. And to convert this number, we
recall that a minute is one sixtieth of a degree. So 𝑑 degrees and 𝑚 minutes is 𝑑
plus 𝑚 divided by 60 degrees. Therefore, 25 degrees and 27
minutes is 25 plus 27 over 60 degrees. And if we calculate this value, we
get 25.45 degrees. We can now update our sketch.
We now want to apply right triangle
trigonometry to this triangle. To do this, we need to start by
labeling the sides. We’ll start with the
hypotenuse. That’s the longest side of the
right triangle, the one opposite the right angle. For this triangle, that’s the side
𝐵𝐶. Next, we see that side 𝐵𝐸 is the
one opposite our known angle of 25.45 degrees. So we’ll label this as the opposite
side. Finally, we can see that side 𝐸𝐶
is the one adjacent to our known angle. So we can label this as the
adjacent side.
To help us recall which
trigonometric ratios we need to use, we’ll use the acronym SOH CAH TOA. Let’s start by determining the
length of the opposite side to our angle. That’s side 𝐵𝐸. To do this, we first note that we
know the value of the hypotenuse and we want to determine the length of the opposite
side. Therefore, we’ll need to use the
sine ratio, which tells us if 𝜃 is an angle in a right triangle, then the sin of 𝜃
will be equal to the length of the side opposite angle 𝜃 divided by the length of
the hypotenuse. Substituting in these values, we
get the sin of 25.45 degrees is equal to the length of 𝐵𝐸 divided by 28. And we can solve for the length of
𝐵𝐸 by multiplying through by 28. We get that the length of 𝐵𝐸 is
28 sin of 25.45 degrees.
But remember, we’re trying to find
the length of the diagonal 𝐵𝐷. And that’s twice the length of
𝐵𝐸. So we can actually multiply both
sides of this equation through by two to determine the length of 𝐵𝐷. This gives us that the length of
𝐵𝐷 is 56 sin of 25.45 degrees. Typing this into our calculator and
making sure it’s set to degrees mode, we get 24.0645 and this expansion continues
meters.
Remember, the question wants us to
give our answer to three decimal places. So we need to look at the fourth
decimal digit, which is five, which is greater than or equal to five. So we need to round this value
up. This gives us 24.065 meters to
three decimal places.
We could now follow the exact same
process to find the length of the other diagonal 𝐴𝐶. We would apply right triangle
trigonometry to 𝐵𝐸𝐶 to find the length of 𝐸𝐶 and then multiply this value by
two. However, this is not the only
method we could use. We now know the length of the
hypotenuse and the length of the opposite side in this right triangle. So we could also use the
Pythagorean theorem. We can recall the Pythagorean
theorem tells us in a right triangle, the square of the hypotenuse is equal to the
sum of the square of the two shorter sides. Rearranging this equation for our
right triangle, we get 𝐸𝐶 squared is equal to 𝐵𝐶 squared minus 𝐵𝐸 squared.
Substituting in our expressions for
the lengths of 𝐵𝐶 and 𝐵𝐸, we get 𝐸𝐶 squared is 28 squared minus 28 times the
sin of 25.45 degrees squared. And we can solve for 𝐸𝐶 by taking
the square root of both sides of the equation. This gives us the length of 𝐸𝐶 is
the square root of 28 squared minus 28 times the sin of 25.45 degrees squared.
We could evaluate this, but
remember we want to find the length of the diagonal 𝐴𝐶. And that’s twice the length of
𝐸𝐶. So we’ll first multiply this
equation through by two. 𝐴𝐶 has a length of two root 28
squared minus 28 times the sin of 25.45 degrees squared. And now we can evaluate this
expression by using our calculator. We get 50.5657 and this expansion
continues meters. Finally, we need to round this
value to three decimal places. We see the fourth decimal digit is
seven. So we need to round this value up,
which gives us the length of 𝐴𝐶 to three decimal places is 50.566 meters.
Therefore, we were able to show if
𝐴𝐵𝐶𝐷 is a rhombus with side lengths 28 meters and the measure of angle 𝐴𝐵𝐶
129 degrees and six minutes, then the length of the diagonal 𝐴𝐶 to three decimal
places is 50.566 meters. And the length of the diagonal 𝐵𝐷
to three decimal places is 24.065 meters.
