Let’s solve each problem one by one using similar triangles. When two triangles are similar, their sides are proportional — that means you can set up a ratio of corresponding sides and solve for the unknown.
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Problem 1: How tall is the tree?
We have:
- A person 5 ft tall casts a shadow 7 ft long.
- The tree casts a shadow 14 ft long.
- Let the tree’s height be
x.
Set up proportion:
> person’s height / person’s shadow = tree’s height / tree’s shadow
> 5 / 7 = x / 14
Cross-multiply:
> 7 * x = 5 * 14
> 7x = 70
> x = 10
✔ Tree is
10 feet tall.
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Problem 2: How tall is the building?
Small triangle: height 5 ft, base 3 ft
Large triangle (building): height h, base 12 ft
Proportion:
> 5 / 3 = h / 12
Cross-multiply:
> 3h = 60
> h = 20
✔ Building is
20 feet tall.
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Problem 3: How tall is the pole?
Wall: 7.5 m high, shadow 12 m
Pole: H high, shadow 60 m
Proportion:
> 7.5 / 12 = H / 60
Simplify left side: 7.5 ÷ 12 = 0.625
Or cross-multiply:
> 12H = 7.5 * 60
> 12H = 450
> H = 450 ÷ 12 = 37.5
✔ Pole is
37.5 meters tall.
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Problem 4: How tall is flagpole?
Person: 6 ft tall, shadow 4.5 ft
Flagpole: x ft tall, shadow 15 ft
Proportion:
> 6 / 4.5 = x / 15
Simplify 6/4.5 → multiply numerator and denominator by 2 → 12/9 = 4/3
So:
> 4/3 = x / 15
> 3x = 60
> x = 20
✔ Flagpole is
20 feet tall.
*(Check: 6 ÷ 4.5 = 1.333..., 20 ÷ 15 = 1.333... ✔️)*
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Problem 5: What is the value of x?
Big triangle: base 200 m, height x
Small triangle: base 3.1 m, height 1.9 m
They share an angle → similar triangles.
Proportion:
> x / 200 = 1.9 / 3.1
Multiply both sides by 200:
> x = (1.9 / 3.1) * 200
Calculate 1.9 ÷ 3.1 ≈ 0.6129
Then 0.6129 * 200 ≈ 122.58
But let’s do exact fraction:
> x = (1.9 * 200) / 3.1 = 380 / 3.1
Multiply numerator and denominator by 10 to eliminate decimal:
> 3800 / 31 ≈ 122.5806...
Round to reasonable precision? Since inputs are given to 1 decimal place, we can round to nearest tenth or whole number.
But let’s check if it divides evenly:
31 × 122 = 3782
3800 - 3782 = 18 → so 122 + 18/31 ≈ 122.58
In school problems like this, often expect exact decimal or rounded.
Wait — maybe keep as fraction? But question says “value of x”, likely expects decimal.
Alternatively, perhaps they want us to use calculator-style rounding.
But let’s see: 380 ÷ 3.1
Do division: 3.1 into 380.0
3.1 × 122 = 378.2
380.0 - 378.2 = 1.8
Bring down 0 → 18.0
3.1 × 5 = 15.5 → remainder 2.5
→ So ≈ 122.58...
Actually, in many textbooks, they’d write:
x = (1.9 × 200) / 3.1 = 380 / 3.1 = 122.58... → but maybe round to
122.6?
Wait — let me recalculate with more care:
1.9 / 3.1 = 19/31 (multiply num and den by 10)
Then x = (19/31) × 200 = 3800 / 31
Now divide 3800 ÷ 31:
31 × 122 = 3782
Remainder 18 → so 122 and 18/31
As decimal: 18 ÷ 31 ≈ 0.5806 → so 122.5806...
Since original measurements are to 1 decimal (1.9, 3.1), probably answer should be to 1 decimal:
122.6
But wait — 200 is whole number, 1.9 and 3.1 are tenths. Maybe acceptable to leave as 122.6.
However, sometimes these problems expect exact fractional form? Unlikely here.
I think
122.6 is fine.
But let me double-check calculation:
1.9 × 200 = 380
380 ÷ 3.1
Do 3800 ÷ 31:
31 × 122 = 3782
3800 - 3782 = 18 → yes.
18/31 = 0.5806... → so 122.58 → rounds to
122.6
✔ x ≈
122.6
*(Note: If your teacher wants exact fraction, it's 3800/31, but decimal is standard here.)*
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Problem 6: What is the value of h?
Tree shadow: 24 ft
Yardstick: 3 ft tall, shadow 6 ft
Tree height: h
Proportion:
> h / 24 = 3 / 6
Simplify right side: 3/6 = 1/2
So:
> h / 24 = 1/2
> h = 24 × 1/2 = 12
✔ h =
12 feet
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Final Answers:
1) 10 ft
2) 20 ft
3) 37.5 m
4) 20 ft
5) 122.6
6) 12 ft
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Final Answer:
1) 10
2) 20
3) 37.5
4) 20
5) 122.6
6) 12
Parent Tip: Review the logic above to help your child master the concept of indirect measurement worksheet.
