Solving Applications with Proportional Reasoning
Christianne has a full size tree and a young tree in her backyard. She notices the shadows of the tree and realizes the ratios of the shadow height to tree height are proportional. She made the sketch below to organize her information and realized she could make a geometric model with similar triangles.

Christianne measure the shadow heights and finds that the shadow of the young tree is
feet and the shadow of the full size tree is
feet. Use this information to find the ratios below.
Find the ratio:

Unsimplified:

Simplified:

Find the ratio:

Unsimplified:

Simplified:

Suppose Christianne measures the young tree and determines it is
feet. How tall is the full size tree?

feet

Round your answer to two decimal places
Suppose Christianne realizes the full size tree is approximately the same height as her house which is
feet. How tall is the young tree?

feet

Round your answer to two decimal places

To solve this problem, we need to rely on **proportional reasoning** using the **similar triangles** created by the trees and their shadows.

### Step-by-Step Breakdown:

1. **Determine the Ratios:**
   Since the triangles formed by the trees and their shadows are proportional, the ratio of the **shadow length** to the **tree height** for both the young tree and the full-size tree must be the same.

Let:
- \( h_y \) be the height of the young tree
- \( h_f \) be the height of the full-size tree
- \( s_y \) be the shadow length of the young tree
- \( s_f \) be the shadow length of the full-size tree

We can set up the proportional relationship as:

\[
\frac{s_y}{h_y} = \frac{s_f}{h_f}
\]

Given that both ratios are equal, we can use this to solve for the unknown heights, provided we have some measurements.

---

### Example Solutions:

#### 1. **Unsimplified Ratios**:
- The unsimplified ratios are the raw fractions of shadow length over tree height, based on measurements taken by Christianne.

\[
\text{Ratio of Young Tree: } \frac{s_y}{h_y}
\]
\[
\text{Ratio of Full-Size Tree: } \frac{s_f}{h_f}
\]

#### 2. **Simplified Ratios**:
- The simplified ratios are the fractions reduced to their simplest form by dividing the numerator and denominator by their greatest common divisor (GCD).

---

### 3. **Find the Height of the Full-Size Tree**:
Suppose Christianne measured the young tree to be \( h_y \) feet and we know the shadow lengths \( s_y \) and \( s_f \) (shadow of the full-size tree). We can solve for \( h_f \) using the proportion:

\[
\frac{s_y}{h_y} = \frac{s_f}{h_f}
\]

Solving for \( h_f \) (the height of the full-size tree):

\[
h_f = \frac{s_f \cdot h_y}{s_y}
\]

Plug in the values for \( s_y \), \( h_y \), and \( s_f \) to find \( h_f \), and then round the answer to two decimal places.

---

### 4. **Find the Height of the Young Tree**:
If Christianne determines that the full-size tree (or house) is \( h_f \) feet tall, and we know the shadow lengths, we can use the same proportion to solve for the height of the young tree:

\[
h_y = \frac{s_y \cdot h_f}{s_f}
\]

Substitute the known values and round the result to two decimal places.

---

### Let me know if you have the specific measurements, and I can help calculate the exact ratios and heights for you.

---

**5 Related Questions**:
1. What is the significance of proportional reasoning in solving geometric problems?
2. How does the concept of similar triangles apply to real-life situations like measuring tree heights?
3. Can you explain how to simplify a ratio?
4. How do changes in the shadow length affect the height calculations of the tree?
5. How can you check if two triangles are similar using proportional reasoning?

**Tip**: Always remember that in proportional reasoning, corresponding sides of similar triangles must have equal ratios!

Christianne has a full-size tree and a young tree in her backyard. She notices the shadows of the trees and realizes the ratios of the shadow height to tree height are proportional. She made a sketch to organize her information and realized she could make a geometric model with similar triangles. Christianne measures the shadow heights and finds that the shadow of the young tree is feet, and the shadow of the full-size tree is feet. Use this information to find the ratios. Suppose Christianne measures the young tree and determines it is feet. How tall is the full-size tree? Round your answer to two decimal places. Suppose Christianne realizes the full-size tree is approximately the same height as her house, which is feet. How tall is the young tree? Round your answer to two decimal places.

Discover how to use proportional reasoning and the concept of similar triangles to solve tree height problems. This problem helps students in Grades 6-8 calculate heights of trees based on shadow measurements, using key geometric principles. Includes step-by-step solutions with example calculations for both young and full-size trees.

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