Math 10 worksheet on similar triangles with word problems involving real-world applications such as calculating heights using shadows and solving for side lengths of similar triangles.

Here are the step-by-step solutions for each problem on the worksheet.

1. Statue Height

Problem: Find the height of the statue ($x$).
* Given: The person is $1.8 \text{ m}$ tall and stands $2.4 \text{ m}$ from the reflection point. The statue is $4 \text{ m}$ from the reflection point.
* Logic: This creates two similar triangles. We can set up a ratio comparing height to distance from the reflection point.
$$ \frac{\text{Statue Height}}{\text{Statue Distance}} = \frac{\text{Person Height}}{\text{Person Distance}} $$
$$ \frac{x}{4} = \frac{1.8}{2.4} $$
* Calculation:
Multiply both sides by 4 to isolate $x$:
$$ x = \frac{1.8 \times 4}{2.4} $$
$$ x = \frac{7.2}{2.4} $$
$$ x = 3 $$

2. Brad's Shadow

Problem: Find the length of Brad's shadow.
* Given: Tree height = $24 \text{ ft}$, Tree shadow = $12 \text{ ft}$. Brad's height = $6 \text{ ft}$.
* Logic: Compare the tree's height to its shadow to find the ratio, then apply it to Brad.
Ratio: $\frac{24}{12} = 2$. The object is twice as tall as its shadow.
* Calculation:
Since Brad is $6 \text{ ft}$ tall, his shadow must be half his height.
$$ \text{Shadow} = \frac{6}{2} = 3 \text{ ft} $$

3. Triangle Sides (EFG and QRS)

Problem: Find the longest side of triangle QRS.
* Given: Triangle EFG sides are $144, 128, 112$. Smallest side of QRS is $280$.
* Logic: First, identify the smallest side of EFG, which is $112$. Now find the scale factor between the smallest sides of the two triangles.
Scale Factor = $\frac{\text{New Side}}{\text{Old Side}} = \frac{280}{112} = 2.5$.
To find the longest side of QRS, multiply the longest side of EFG ($144$) by this scale factor.
* Calculation:
$$ 144 \times 2.5 = 360 $$

4. Building Shadow

Problem: Find the shadow cast by a $200 \text{ ft}$ building.
* Given: Flagpole height = $40 \text{ ft}$, Flagpole shadow = $25 \text{ ft}$. Building height = $200 \text{ ft}$.
* Logic: Set up a proportion: $\frac{\text{Flagpole Height}}{\text{Flagpole Shadow}} = \frac{\text{Building Height}}{\text{Building Shadow}}$.
$$ \frac{40}{25} = \frac{200}{x} $$
* Calculation:
Cross-multiply: $40x = 25 \times 200$
$40x = 5000$
Divide by 40: $x = \frac{5000}{40} = 125$

5. Lamp Post Height

Problem: How high is the lamp post?
* Given: Girl height = $160 \text{ cm}$. She stands $360 \text{ cm}$ from the post. Her shadow is $90 \text{ cm}$ long.
* Logic: There are two similar triangles here.
1. Small triangle: Base is the shadow ($90 \text{ cm}$), Height is the girl ($160 \text{ cm}$).
2. Large triangle: Base is the total distance from the light source to the tip of the shadow ($360 + 90 = 450 \text{ cm}$). Height is the lamp post ($h$).
Ratio: $\frac{\text{Girl}}{\text{Shadow}} = \frac{\text{Lamp}}{\text{Total Distance}}$
$$ \frac{160}{90} = \frac{h}{450} $$
* Calculation:
Simplify $\frac{160}{90}$ to $\frac{16}{9}$.
$$ \frac{16}{9} = \frac{h}{450} $$
Multiply by 450: $h = \frac{16 \times 450}{9}$
$450 / 9 = 50$.
$16 \times 50 = 800 \text{ cm}$.

6. Tower Height

Problem: How high is the tower?
* Given: Stick height = $1.2 \text{ m}$, Stick shadow = $0.6 \text{ m}$. Tower shadow = $7 \text{ m}$.
* Logic: Notice that the stick ($1.2$) is exactly double the length of its shadow ($0.6$). Therefore, the tower must also be double the length of its shadow.
* Calculation:
$$ \text{Tower Height} = 7 \times 2 = 14 \text{ m} $$

7. Triangle Perimeter (IJK and TUV)

Problem: What is the perimeter of TUV?
* Given: IJK sides are $40, 50, 24$. Longest side of TUV is $275$.
* Logic: Identify the longest side of IJK, which is $50$. Find the scale factor using the longest sides.
Scale Factor = $\frac{275}{50} = 5.5$.
Calculate the perimeter of IJK first: $40 + 50 + 24 = 114$.
Multiply the perimeter of IJK by the scale factor to get the perimeter of TUV.
* Calculation:
$$ 114 \times 5.5 = 627 $$

8. Second Tree Height

Problem: How high is the second tree?
* Given: Tree 1 height = $4 \text{ m}$, shadow = $15 \text{ m}$. Tree 2 shadow = $20 \text{ m}$.
* Logic: Set up a proportion: $\frac{\text{Height}_1}{\text{Shadow}_1} = \frac{\text{Height}_2}{\text{Shadow}_2}$.
$$ \frac{4}{15} = \frac{h}{20} $$
* Calculation:
Cross-multiply: $15h = 4 \times 20$
$15h = 80$
$h = \frac{80}{15}$
Simplify fraction (divide by 5): $\frac{16}{3}$ or approx $5.33 \text{ m}$.

9. Triangle Perimeter (CDE and NOP)

Problem: What is the perimeter of NOP?
* Given: Perimeter of CDE = $133$. Corresponding sides are $53$ (smaller) and $212$ (larger).
* Logic: Find the scale factor between the corresponding sides.
Scale Factor = $\frac{212}{53} = 4$.
Multiply the perimeter of the smaller triangle by the scale factor.
* Calculation:
$$ 133 \times 4 = 532 $$

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Final Answer:
1. 3 m
2. 3 feet
3. 360
4. 125 feet
5. 800 cm
6. 14 m
7. 627
8. 5.33 m (or 16/3 m)
9. 532