# Find the distance between points C and D if the height of the tree is 4m

• MHB
• Elissa89
In summary, using the given angles of elevation and the height of the tree, we can use trigonometry to find the distance between points C and D. By setting up equations using tangent and cotangent, we can solve for the distances and then find the difference between them to get the total distance.
Elissa89
So the question is...From point C on the ground level, the angle of elevation to the top of a tree is 30 degrees. From point D, which is closer to the tree, the angle of elevation is measured to be 45 degrees. Find the distance between points C and D if the height of the tree is 4m.

I know triangle 1 has angles 30 degrees, 60 and 90. So the adjacent side is 4*sqrt(3)

I know triangle 2 has angles 45 degrees, 45 and 90. The adjacent side is 2*sqrt(2)

From here I am stuck as I do not know how to find the distance between point C and D

I would let $$d_C$$ be the distance from point C to the tree in meters, and so we may write:

$$\displaystyle \tan\left(30^{\circ}\right)=\frac{4}{d_C}\implies d_C=4\cot\left(30^{\circ}\right)=4\sqrt{3}\quad\checkmark$$

Likewise for point D:

$$\displaystyle \tan\left(45^{\circ}\right)=\frac{4}{d_D}\implies d_D=4\cot\left(45^{\circ}\right)=4$$

We should expect that in a 45-45-90 triangle the adjacent and opposite sides are equal. And so the distance $$d$$ between the two points is:

$$\displaystyle d=d_C-d_D=?$$

## What is the formula for finding the distance between two points?

The formula for finding the distance between two points is the square root of (x2-x1)^2 + (y2-y1)^2. This is also known as the Pythagorean theorem.

## How do I determine the coordinates of points C and D?

In order to determine the coordinates of points C and D, you will need to know the location of the tree and the distance from the tree to points C and D. The coordinates can then be calculated using the distance formula.

## Why is the height of the tree important in finding the distance between points C and D?

The height of the tree is important because it gives us a reference point for the distance between points C and D. Without the height, we would only have two-dimensional coordinates and would not be able to accurately determine the distance.

## Can the distance between points C and D be negative?

No, the distance between two points cannot be negative. Distance is always a positive value, representing the length between two points.

## How can I use the distance between points C and D in my research or experiments?

The distance between points C and D can be used in various ways in scientific research and experiments. For example, it can help determine the size or location of objects, measure the effects of certain factors on distance, or aid in creating accurate models or simulations. It can also be used to calculate velocity, acceleration, and other important variables in physics and engineering.

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