Brian McCall

**Univ. of Wisconsin**

J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

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Indirect measurement is a method of using proportions to find an unknown length or distance in similar figures. Two common ways to achieve **indirect measurement** involve (1) using a mirror on the ground and (2) using shadow lengths and find an object's height. Method 1 measures the person's height and the distances between the person, mirror, and object. Method 2 measures shadows and the person's height.

You can use similar triangles to measure something without actually taking a tape measure and stretching it out. There are 2 methods.

One method is if you use a mirror. So if you as a person, put a mirror on the ground. So I'm going to label that this is a mirror and if you backed up all the way until you could just see the very tip of whatever you're trying to measure. So if my eyes are right here, then I'm going to use a special property of mirrors and light. And what that is is that light, bouncing off a mirror will bounce off at a similar angle. And by similar I mean congruent. So what we've done is we've created congruent angles there. So again what I did is I'm backed all the way up, I kept the mirror in place or I guess I could stay in place and have someone move the mirror until I could see just very tippy top of whatever I'm trying to measure.

The way that we create similar triangles here is assuming I'm standing perfectly straight. So this is a right angle. And assuming that nature did a good job of growing this tree and this is a right angle, then we have an angle angle shortcut so we have 2 similar triangles which I could draw down below. So that will be your right angle and then this angle would be congruent to the angle formed by light bouncing off that mirror. So you could use my height which you can measure, you can measure the distance between me and the mirror, you can measure the distance between the mirror and your object, and you could use that proportion to find that missing side, the height of your object. So that's one way of using similar triangles to measure something tall.

The second method uses shadows. So let's say you had the sun up here and at the same time you measured a person's shadow and a shadow of whatever you're trying to measure. So let's say it was a tree. So here is the sun and let's say at some point in time I cast a shadow like that and then the object I'm trying to measure in this case a tree cast a shadow which will be a little bit longer. Then what I can do is I can basically create some more triangles. Assuming that the sun angle does not change, that is if I measure these at the exact same time, then I will create a 90 degree angle here, a 90 degree angle here, and this angle between myself and the angle of the sun will be constant between me and the object I'm trying to measure. So in this case you would measure the shadow length and the person's height and then over here you would measure the shadow length. And you could use similar triangles to find the height of your object.

Both methods work and this one will always work and this one would work if you have a mirror handy.