Thread: 4 Triangle Questions

For 1, I know the definitions of Similar and congruent; but couldnt find the sets of triangles on google. 2. Tried using google, but didnt find anything helpfull. # 3 & 4 I forgot how to do it, a similar example would be great.


1.Explain what is meant by similar and congruence. Describe the conditions for each set of triangles.

2.How is the area of two similar triangles related to the length of the sides?


3.While on a train journey through northern Ontario, a passenger was curious about the height of the train. At one of the stops, she noticed that a passenger was standing next to the train. She knew his height, and knew she could measure the length of his shadow and that of the train's. The passenger was 1.8 m tall, casting a shadow 0.6 m long. The shadow of the train was 2.4 m long. Use the principles of similar triangles to find the height of the train. Draw a sketch and show your calculations.


4.Brian decided to help his parents with their triangular garden (which measures 3 m by 4 m by 5 m). They want to extend the garden, but keep it in the shape of a triangle. They would like to extend the 3 m and 4 m sides by a factor of 1.5. His parents are concerned about the area of the new garden, since they have only so many flowers to plant. Brian assures them that the entire area of the new garden won't be that big. He is able to use the principles of similar triangles to show his parents. Describe the solution Brian could present to his parents. Draw a sketch and show your solution.

Originally Posted by Serialkisser

1.Explain what is meant by similar and congruence. Describe the conditions for each set of triangles.

I don't think I can give a better explanation of similar than this page does:

Similar Triangles

Roughly speaking, two triangles are similar if one is a scaled copy of the other.

Any two congruent triangles are similar, but the corresponding sides of two congruent triangles are of equal length. That is, the triangles are similar but one is a scaled copy of the other by a factor of exactly 1.

Originally Posted by Serialkisser

3.While on a train journey through northern Ontario, a passenger was curious about the height of the train. At one of the stops, she noticed that a passenger was standing next to the train. She knew his height, and knew she could measure the length of his shadow and that of the train's. The passenger was 1.8 m tall, casting a shadow 0.6 m long. The shadow of the train was 2.4 m long. Use the principles of similar triangles to find the height of the train. Draw a sketch and show your calculations.

I'm assuming that both the train and the passenger are at right angles to the ground.

The passenger is one side of a triangle, his shadow is another, and a line from the top of his head to the tip of his shadow is the third. The angle between the passenger and the line to the tip of his shadow is determined by the sun. We don't know it, but it will be the same for the train, so we can call it . Since we have two angles (the right angle between the passenger and his shadow) and , we can subtract them both from 180 to find the third angle.

Now, the train is also at a right angle to its shadow, and the position of the sun also forms the same angle between the train and the line from the top of the train to the tip of its shadow. Just as with the passenger, we can find the third angle by subtracting the right angle and from 180.

Now, the actual numeric values of the angles aren't important, but the fact that they are the same between the passenger-triangle and the train-triangle is the key. This shows that the triangles are similar.

Since similar triangles are just scaled copies of each other, you can scale each side by the same factor.

For example, suppose the train's shadow was 3m long and the man's shadow was only 1m. Then the train-triangle would be exactly three times the size of the passenger triangle.

Thanks to the support I got; I was able to do the other questions; except these two.

While on a train journey through northern Ontario, a passenger was curious about the height of the train. At one of the stops, she noticed that a passenger was standing next to the train. She knew his height, and knew she could measure the length of his shadow and that of the train's. The passenger was 1.8 m tall, casting a shadow 0.6 m long. The shadow of the train was 2.4 m long. Use the principles of similar triangles to find the height of the train. Draw a sketch and show your calculations.


I tried and tried, but I dont get it; a right angle you said:

So the triangle I made is:
A-B 1.8m Person
B-C 0.6m Shadow
A-C 2.4m Shadow of train
I dont know how to move on, and im also unsure if my triangle is correkt



Brian decided to help his parents with their triangular garden (which measures 3 m by 4 m by 5 m). They want to extend the garden, but keep it in the shape of a triangle. They would like to extend the 3 m and 4 m sides by a factor of 1.5. His parents are concerned about the area of the new garden, since they have only so many flowers to plant. Brian assures them that the entire area of the new garden won't be that big. He is able to use the principles of similar triangles to show his parents. Describe the solution Brian could present to his parents. Draw a sketch and show your solution.

Thanks,

Serialkisser

I just did number 3; the train problem. I just need someone who could check if my answer is right. I attached the Image with this post.

Brian decided to help his parents with their triangular garden (which measures 3 m by 4 m by 5 m). They want to extend the garden, but keep it in the shape of a triangle. They would like to extend the 3 m and 4 m sides by a factor of 1.5. His parents are concerned about the area of the new garden, since they have only so many flowers to plant. Brian assures them that the entire area of the new garden won't be that big. He is able to use the principles of similar triangles to show his parents. Describe the solution Brian could present to his parents. Draw a sketch and show your solution.


I drew 2 similar triangles, oe with the given mesurments. For the 2nd Triangle, I multiplied the factor 1.5 by the sides Brian wants to extend. How do I move on ?
The scale factor of the triangles' sides is 1.5. I know that in similar triangles, the ratio of the areas is the square of the scale factor. But I dont think a short answer like this would be accepted. I dont know how to describe it, and how to get there :S.

Originally Posted by Serialkisser

I just did number 3; the train problem. I just need someone who could check if my answer is right. I attached the Image with this post.

Sorry it took so long to get back to you. You have the setup correct for the train problem, but it appears you've made an error in the algebra on the second step:





Instead, it should be:

Originally Posted by Serialkisser

I know that in similar triangles, the ratio of the areas is the square of the scale factor. But I dont think a short answer like this would be accepted. I dont know how to describe it, and how to get there :S.

The formula for the area of a triangle is , where is the triangle's base and is the triangle's height.

Here is a page that gives a pretty good introduction to triangle areas: Area of a Triangle

If you scale a triangle, you're scaling both the height and the base. Let capital letters stand for the new base and height, like this:




Then if the old area was , then the new area would be . If you substitute in the expressions using the original variables and , you get:



In Brian's problem, this method can be used to find the area of the expanded garden:



Let me know if anything is unclear. Good luck with the triangles!