often times you can use similar triangles in the real world. sarah is attempting to find the height of a tree by using a mirror. by the laws of physics she knows that the angle of incidence is congruent to the angle of reflection (these are marked red on the diagram). explain why the two triangles are similar and if she knows her height and the distance from the mirror to her feet and also the distance from the mirror to the tree how could she calculate the height of the tree.

The two triangles are similar because they have two angles in common (by the AA theorem).

The height of the tree can be calculated by figuring out the ratio between the distance between the mirror to her feet and the distance from the mirror to the tree, then applying this to her height and the height of the tree (x).

Step-by-step explanation:

By the Law of Reflection, the angle of incidence (the ray of light that hits the mirror first) and the angle of reflection (the ray of light that is reflected off the mirror) are equal to each other.

Triangles can be proved similar by the AA, SAS, or SSS theorems. In this particular case, the triangles can be proved similar by the AA theorem.

We know that both triangles have one congruent angle in common.

Sarah is standing straight, you can say perpendicular to the ground. The tree is also standing straight, also perpendicular to the ground.

Therefore, we can conclude that the angles formed by Sarah and the tree are right angles. Now this means that the two triangles have two angles in common, making them similar triangles by the AA (Angle Angle) theorem.

Since the triangles are similar, the ratios of the sides of the triangles will be the same. (For example, if side AB on Triangle 1 was 4 and side DE on Triangle 2 was 8, the ratio would be 1:2)

So if Sarah knows the distance from the mirror to her feet and the distance from the mirror to the tree, she can create the ratio between the two triangles.

Knowing this ratio, she can take her height and use the ratio she figured out to determine the height of the tree.  

If this was still unclear to you, I highly recommend reading the Example!

Say the distance from the mirror to her feet was 3 feet and the distance from the mirror to the tree was 9 feet. This means that the ratio would be 1:3 (treat the numbers as a fraction; simplify the fraction).

Knowing the ratio, if her height was 5 feet then the tree would be 3 times her height (ratio 1:3).

How I figured that out was by putting 5 feet into "1" and since I had to multiply 1 by 5, I would also have to multiply "3" by 5, giving me 15.

I hope this answer helped! I attached an image showing the numbers I worked with. Leave any questions in the comments.

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step-by-step explanation: