Question

Answer in complete and through sentences.

You are on a jungle expedition and come to a raging river. You need to build a bridge across the river. You spot a tall tree directly across from you on the opposite bank (point A). You place a pole in the ground to mark the position directly across from the tree. From the spot you are standing, you walk directly downstream 16 feet and place another pole (point C). You keep walking down stream and mark a third spot with a pole that is 7 feet from your last pole (point D). Turning perpendicular from the last pole, you walk away from the river 9 feet to another position and place a fourth pole (point E).


a. Determine if the two triangles are similar. Explain which similarity theorem you used and why.
b. Use proportions to calculate the distance across the river.

A large tree is close to point C and we could chop it down at an angle to reach point A and safely cross the river.

c. How tall does the tree need to be to span the river from point C to A?

Answer 1

To determine if the two triangles are similar, we can compare their corresponding angles and side lengths.Let's label the points:The point across the river from your initial position is labeled as B.The point downstream where you placed the second pole is labeled as C.The point downstream where you placed the third pole is labeled as D.The point where you walked away from the river and placed the fourth pole is labeled as E.Now, let's compare the corresponding angles of the two triangles:In triangle ABC:Angle ABC is a right angle (90 degrees) since it's formed by a line perpendicular to the river and the river bank.Angle BAC is the angle you are standing at, which we'll call angle θ.In triangle CDE:Angle CDE is a right angle (90 degrees) since it's formed by a line perpendicular to the river and the river bank.Angle DCE is the same angle as angle θ since they are alternate interior angles (formed by a transversal intersecting two parallel lines).

So far, we can see that the two triangles have one pair of congruent angles (90 degrees) and one pair of corresponding angles (θ and DCE) that are equal.Now, let's compare the corresponding side lengths of the two triangles:In triangle ABC:Side AB is the distance across the river that we need to calculate.Side BC is the distance you walked downstream, which is 16 feet.In triangle CDE:Side CE is the distance you walked away from the river, which is 9 feet.Side DE is the distance between the third and fourth poles, which is 7 feet.We can see that the ratios of the corresponding side lengths are:AB/BC = ?/16

CE/DE = 9/7To determine if the triangles are similar, we need the ratios of the corresponding side lengths to be equal. Let's solve for the unknown side length, AB:AB/BC = CE/DEAB/16 = 9/7Cross-multiplying, we have:AB * 7 = 16 * 9AB = (16 * 9) / 7AB ≈ 20.57 feetSo, the distance across the river (side AB) is approximately 20.57 feet.In conclusion:

a. The two triangles, ABC and CDE, are similar because they have one pair of congruent angles (90 degrees) and one pair of corresponding angles (θ and DCE) that are equal. This similarity is based on the Angle-Angle (AA) similarity theorem.

b. The distance across the river (side AB) is approximately 20.57 feet, calculated using proportions based on the similarity of the triangles.

Explanation:

Hope this helps!

Answer 2

To determine if the two triangles are similar, we can compare their corresponding angles and side lengths.

Let's compare the corresponding angles of the two triangles:

So, in triangle ABC:

Angle ABC is a right angle (90 degrees) since it's formed by a line perpendicular to the river and the river bank. Angle BAC is the angle we are standing at, which we'll call angle 1.

In triangle CDE: Angle CDE is a right angle (90 degrees) since it's formed by a line perpendicular to the river and the river bank. Angle DCE is the same angle as angle 1 since they are alternate interior angles (which is formed by a transversal intersecting two parallel lines). Right now, we can see that the two triangles have ONE pair of congruent angles (90 degrees) and one pair of corresponding angles (Angle 1 and DCE) that are equal.

Now, let's compare the corresponding side lengths of the two triangles:

In triangle ABC:

Side AB is the distance across the river that we need to calculate. Side BC is the distance you walked downstream, which is 16 feet.

In triangle CDE:

Side CE is the distance you walked away from the river, which is 9 feet. Side DE is the distance between the third and fourth poles, which is 7 feet.

We can see that the ratios of the corresponding side lengths are:

`AB/BC = CE/DE \\AB/16 \\9/7`

To know if the triangles are similar, we need the ratios of the corresponding side lengths to be equal. To solve for the unknown side length, we need to do AB:

`AB/BC = x/16`

`CE/DE = 9/7`

Cross-multiplying, we have: AB * 7 = 16 * 9AB = (16 * 9) / 7AB ≈ 20.57 feet. So, the distance across the river (side AB) is approximately 20.57 feet. In conclusion:

1. The two triangles, ABC and CDE, are similar because they have one pair of congruent angles (90 degrees) and one pair of corresponding angles (1 and DCE) that are equal.

This similarity is based on the Angle-Angle (AA) similarity theorem.

2. The distance across the river (side AB) is approximately 20.57 feet, calculated using proportions based on the similarity of the triangles.

I hope this helps :)