Question

1. the width of a rectangle is 2 m less than the length. the area is 48 m². find the dimensions.
2. one side of a right triangle is 2 cm shorter than the hypotenuse and 7 cm longer than the third side. find the lengths of the sides of the triangle.

Explanation:

Problem 7: Rectangle Dimensions

Step1: Define variables

Let length = \( l \) m, width = \( l - 2 \) m (since width is 2 m less than length).

Step2: Area formula for rectangle

Area of rectangle = \( \text{length} \times \text{width} \). Given area = 48 m², so:
\( l(l - 2) = 48 \)

Step3: Solve quadratic equation

Expand: \( l^2 - 2l - 48 = 0 \)
Factor: \( (l - 8)(l + 6) = 0 \)
Solutions: \( l = 8 \) (since length can’t be negative, \( l = -6 \) is discarded).

Step4: Find width

Width = \( 8 - 2 = 6 \) m.

Problem 8: Right Triangle Sides

Step1: Define variables

Let hypotenuse = \( h \) cm, one leg = \( h - 2 \) cm (2 cm shorter than hypotenuse), other leg = \( (h - 2) - 7 = h - 9 \) cm (7 cm longer than the third side? Wait, re-express: Let third side (leg) = \( x \) cm. Then one leg = \( x + 7 \) cm, hypotenuse = \( (x + 7) + 2 = x + 9 \) cm.

Step2: Pythagorean theorem

\( x^2 + (x + 7)^2 = (x + 9)^2 \)

Step3: Expand and simplify

\( x^2 + x^2 + 14x + 49 = x^2 + 18x + 81 \)
\( 2x^2 + 14x + 49 - x^2 - 18x - 81 = 0 \)
\( x^2 - 4x - 32 = 0 \)

Step4: Solve quadratic

Factor: \( (x - 8)(x + 4) = 0 \)
Solutions: \( x = 8 \) (since length can’t be negative, \( x = -4 \) is discarded).

Step5: Find sides

Third side (leg) = 8 cm, one leg = \( 8 + 7 = 15 \) cm, hypotenuse = \( 8 + 9 = 17 \) cm.

Problem (Middle, Consecutive Even Integers)

Step1: Define integers

Let first even integer = \( 2n \), next = \( 2n + 2 \) (consecutive even integers differ by 2).

Step2: Sum of squares

\( (2n)^2 + (2n + 2)^2 = 452 \)

Step3: Expand and simplify

\( 4n^2 + 4n^2 + 8n + 4 = 452 \)
\( 8n^2 + 8n - 448 = 0 \)
Divide by 8: \( n^2 + n - 56 = 0 \)

Step4: Solve quadratic

Factor: \( (n + 8)(n - 7) = 0 \)
Solutions: \( n = 7 \) (since \( n = -8 \) gives negative integer, discard).

Step5: Find integers

First integer: \( 2(7) = 14 \), next: \( 14 + 2 = 16 \).

Answer:

s:

1. Length = 8 m, Width = 6 m
2. Sides: 8 cm, 15 cm, 17 cm

Consecutive Even Integers: 14, 16