Problem number 28 states, two plane mirrors OP and OQ are inclined at an acute angle. The diagram is not to scale. A ray of light XY parallel to QO strikes mirror OP at Y. The ray is reflected and hits mirror OQ, is reflected again and hits mirror OP and is reflected for a third time and strikes mirror OQ at a right angle at R as shown. The distance OR is 5 centimeters. The ray XY is D centimeters from the mirror OQ. What is the value of D? So this is the diagram of the two mirrors with the light bouncing between them. So the problem tells us that these two angles are right angles, and it also tells us that these two lines are parallel. So if the angle between the two mirrors is alpha, then angle x, y, p is also going to have a value of alpha. Now, the way a mirror works is by reflecting rays off its surface at the same angle they came in, which means that this angle would also be alpha. Let's do the same for these two angles at the bottom. We will call them gamma. Now let's consider this 180 degree angle. It is equal to two copies of gamma and one 90 degree angle. Solving for gamma, we get 45 degrees. Since one of the angles in the right angle triangle is 45 degrees, then so must the other to make a 45-45-90 triangle. Now let's consider these two similar triangles. We know that they are similar because two of their angles are the same, which means that the third is also the same. This allows us to set up a proportion, which reads that OR is to MR as YN is to MN. Here we can substitute the value of OR with 5, as that is the value that we know. Now let's try finding the value of YN. Since YN is the hypotenuse of a 45-45-90 triangle, we know the length of the hypotenuse is root 2 times the length of one of its legs, so D times root 2. So we can substitute that in for our relation. Now we can almost solve, but we still have two unknown variables that might be hard to solve for. Instead, let's just cancel them out, so let's put MN in terms of MR. We can do this by realizing that MN is the hypotenuse of the 45-45-90 triangle MNR, and that value of the hypotenuse is root 2 times that of the side length. So that is our final proportion, and now we can solve for D. So let's cancel out the MRs in the denominator, and let's cancel out the root 2s, and we get that D is equal to 5. So the question asked us, what is the value of D? The answer is 5, letter C.

plane mirrors

ray of light

45-45-90 triangles

distance calculation

geometry problem