Geometry and Digit Problems: Solving Rectangle, Right Triangle, and Number Theory Challenges

Math Problem Statement

SHOW EACH A VERIFICATION

GEOMETRY-RELATED PROBLEMS

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

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.
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.
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.

Solution

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

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.

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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Math Problem Analysis

Mathematical Concepts

Geometry
Pythagorean Theorem
Algebra
Number Theory

Formulas

Pythagorean theorem: a^2 + b^2 = c^2
Area of a triangle: (1/2) * base * height
Digit relationships and algebraic equations for two-digit and three-digit numbers

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 8-10

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