A tree casts a shadow that is 26 feet in length. Kayla is standing at the end of the shadow and estimates the distance from her to the top of the tree is 45 feet. How tall is the tree? Show steps to solve this equation?

This problem involves using the **Pythagorean Theorem** because the situation forms a right triangle. The tree's height, the length of the shadow, and the line from Kayla to the top of the tree can be thought of as the sides of a right triangle.

### Let’s break it down:

1. The length of the shadow (base of the right triangle) is 26 feet.
2. The distance from Kayla to the top of the tree (the hypotenuse of the triangle) is 45 feet.
3. The height of the tree is what we are trying to find (the perpendicular side of the right triangle).

### Step 1: Apply the Pythagorean Theorem

The Pythagorean Theorem states:
\[
a^2 + b^2 = c^2
\]
Where:
- \( a \) is the height of the tree (unknown),
- \( b \) is the length of the shadow (26 feet),
- \( c \) is the distance from Kayla to the top of the tree (45 feet).

So, plugging in the values:
\[
a^2 + 26^2 = 45^2
\]

### Step 2: Square the known values
\[
a^2 + 676 = 2025
\]

### Step 3: Isolate \( a^2 \)
\[
a^2 = 2025 - 676
\]
\[
a^2 = 1349
\]

### Step 4: Take the square root of both sides
\[
a = \sqrt{1349}
\]
\[
a \approx 36.74 \, \text{feet}
\]

### Conclusion:
The height of the tree is approximately **36.74 feet**.

---
Would you like more details or have any questions?

Here are 5 related questions:
1. How can you use the Pythagorean Theorem to find the diagonal of a rectangle?
2. How does the Pythagorean Theorem apply to real-world problems like ladders leaning against a wall?
3. How would the height change if the shadow were shorter but the distance to the top remained the same?
4. Can you use trigonometry to solve the same problem if an angle was given?
5. What other methods can be used to estimate the height of a tree?

**Tip:** The Pythagorean Theorem only applies to right-angled triangles, so make sure the situation forms a right triangle before using it!

A tree casts a shadow that is 26 feet in length. Kayla is standing at the end of the shadow and estimates the distance from her to the top of the tree is 45 feet. How tall is the tree? Show steps to solve this equation.

Learn how to find the height of a tree casting a 26 ft shadow, using the Pythagorean Theorem. With a distance of 45 ft from the end of the shadow to the top of the tree, this detailed step-by-step solution shows how to solve this real-world geometry problem. Suitable for Grades 8-10.

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