Question Video: Using Right-Angled Triangle Trigonometry to Find Lengths in Rhombuses Mathematics

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.