Given that the triangles shown are similar, determine 𝑥.
If two polygons are similar, then their corresponding angles are congruent which means they’re the exact same measure, and the measure of their corresponding sides are proportional.
In our diagram, we can see that angle 𝐶 and angle 𝑀 are congruent. Angle 𝐴 and Angle 𝐿 are congruent. And then the remaining angles, Angle 𝐵 and Angle 𝑁, they are congruent. We know this based on the markings of the angles.
From the markings of the angles, we can also tell which sides are proportional. So side 𝐴𝐶 is proportional to side 𝐿𝑀. And side 𝐴𝐵 is proportional to 𝐿𝑁.
This means we can set up a proportion. So 𝐴𝐶 is proportional to 𝐿𝑀, and 𝐴𝐵 is proportional to 𝐿𝑁. So the sides on the numerators are from triangle 𝐴𝐵𝐶, and the sides on the denominators are from triangle 𝐿𝑀𝑁. Now we can plug in our values. 𝐴𝐶 is ten, 𝐿𝑀 is 𝑥, 𝐴𝐵 is nine, and 𝐿𝑁 is sixteen. Now we will find the cross product to solve for 𝑥.
So now we have nine times 𝑥 equals ten times sixteen. Let’s multiply. So nine 𝑥 equals one hundred and sixty. Now divide both sides by nine, and 𝑥 is equal to one hundred and sixty ninths.
Therefore, 𝐿𝑀 equals one hundred and sixty ninths.