SHOW EACH A VERIFICATION

GEOMETRY-RELATED PROBLEMS

1. The width of a rectangle is 9 cm. The length is 1 cm shorter than the diagonal. Find the length of the diagonal.

2. The hypotenuse of a right triangle is 25 cm longer than one leg and 32 cm longer than the other leg. Find the area of the triangle.

3. A vertical pole was broken by the wind. The upper part, still attached, reached a point on the level ground 15 feet from the base. If the upper part is 9 feet longer than the lower part, how tall was the pole?

DIGIT PROBLEMS

1. In a two-digit number, the unit's digit is 3 greater than the tens digit. Find the number if it is 4 times as large as the sum of its digits.
 
2. The sum of the digits of a three-digit number is 14. The hundreds digit being 4 times the units digit. If 594 is subtracted from the number, the order of the digits will be reversed. Find the number.

3. A number is less than 100 and its tens digit is 2 more than its units digit. If the number with the digits reversed is subtracted from the original number, the remainder is 3 times the sum of the digits. Find the number.

Let's solve each problem step-by-step and verify each solution carefully.

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### Geometry Problems

#### Problem 1
*The width of a rectangle is 9 cm. The length is 1 cm shorter than the diagonal. Find the length of the diagonal.*

Let:
- Width = \( 9 \) cm
- Length = \( x \) cm
- Diagonal = \( x + 1 \) cm

Using the Pythagorean theorem, we have:
\[
x^2 + 9^2 = (x + 1)^2
\]
Expanding and simplifying:
\[
x^2 + 81 = x^2 + 2x + 1
\]
\[
81 = 2x + 1
\]
\[
80 = 2x
\]
\[
x = 40
\]
Thus, the diagonal length is \( x + 1 = 40 + 1 = 41 \) cm.

**Answer:** The length of the diagonal is **41 cm**.

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#### Problem 2
*The hypotenuse of a right triangle is 25 cm longer than one leg and 32 cm longer than the other leg. Find the area of the triangle.*

Let:
- One leg = \( a \) cm
- Other leg = \( b \) cm
- Hypotenuse = \( c = a + 25 = b + 32 \)

Since \( c = a + 25 \) and \( c = b + 32 \), equating these two:
\[
a + 25 = b + 32
\]
\[
a - b = 7
\]

We also have, by the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
Substitute \( c = a + 25 \):
\[
a^2 + b^2 = (a + 25)^2
\]
Expanding:
\[
a^2 + b^2 = a^2 + 50a + 625
\]
\[
b^2 = 50a + 625
\]

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SHOW EACH A VERIFICATION GEOMETRY-RELATED PROBLEMS
1. The width of a rectangle is 9 cm. The length is 1 cm shorter than the diagonal. Find the length of the diagonal.

2. The hypotenuse of a right triangle is 25 cm longer than one leg and 32 cm longer than the other leg. Find the area of the triangle.

3. A vertical pole was broken by the wind. The upper part, still attached, reached a point on the level ground 15 feet from the base. If the upper part is 9 feet longer than the lower part, how tall was the pole?

DIGIT PROBLEMS
1. In a two-digit number, the unit's digit is 3 greater than the tens digit. Find the number if it is 4 times as large as the sum of its digits.

2. The sum of the digits of a three-digit number is 14. The hundreds digit being 4 times the units digit. If 594 is subtracted from the number, the order of the digits will be reversed. Find the number.

3. A number is less than 100 and its tens digit is 2 more than its units digit. If the number with the digits reversed is subtracted from the original number, the remainder is 3 times the sum of the digits. Find the number.

Explore solutions for geometry-related and digit problems, including finding the diagonal of a rectangle, calculating the area of a right triangle, and solving number puzzles with digit relationships. Step-by-step solutions and key concepts covered.

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