Triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are
similar. What is the length of 𝐸𝐹?
We know that in any similar
triangles, the corresponding angles are equal. In this case, angle 𝐴 is equal to
angle 𝐷, angle 𝐵 equals angle 𝐸, and angle 𝐶 is equal to angle 𝐹. The three angles in any triangle
sum to 180 degrees. This means that from triangle
𝐴𝐵𝐶, three 𝑥 plus two 𝑥 plus 𝑥 is equal to 180. Simplifying the left-hand side
gives us six 𝑥. We can then divide both sides of
this by six, giving us a value of 𝑥 equal to 30.
The three angles in triangle 𝐴𝐵𝐶
are 90 degrees, 60 degrees, and 30 degrees. This means that the corresponding
angles in triangle 𝐷𝐸𝐹 will also be 90 degrees, 60 degrees, and 30 degrees. This means that triangle 𝐷𝐸𝐹 is
a right-angled triangle. And we can use right angle
trigonometry and our knowledge of special trig angles to calculate the length
We recall the acronym SOHCAHTOA,
which helps us find the sine, cosine, and tangent ratios in right-angled
triangles. The longest side of a right-angled
triangle is opposite the right angle. This is known as the
hypotenuse. If we focus on the 30-degree angle,
we see that length 𝐷𝐸 is opposite this. The length 𝐷𝐹 is adjacent or next
to the 30-degree angle and the right angle.
As we are dealing with the opposite
and hypotenuse, we will use the sine ratio. This states that sin of 𝜃 is equal
to the opposite over the hypotenuse. Substituting in our values gives us
sin of 30 is equal to 7.8 over 𝑥. sin of 30 degrees is one of our special trig
angles. It is equal to one-half. This means that one-half is equal
to 7.8 over 𝑥.
Cross Multiplying at this stage
means we are multiplying both sides by 𝑥 and by two. This gives us 𝑥 is equal to 7.8
multiplied by two. This, in turn, is equal to
15.6. The length 𝐸𝐹 is equal to 15.6
centimeters. It is worth noting here that this
triangle follows a general pattern. If we have a right-angled triangle
where another angle is equal to 30 degrees, then the opposite is always half the
length of the hypotenuse.