Problem 43

Question

The perimeter of a rectangle is $70,$ and its diagonal is $25 .$ Find its length and width.

Step-by-Step Solution

Verified

Answer

The length is 20 and the width is 15, or vice versa (interchangeable).

1Step 1: Understanding the Problem

We need to find the length and width of a rectangle given that its perimeter is 70 and its diagonal is 25. We will use the formulas for the perimeter and the Pythagorean theorem to find the relationship between length and width.

2Step 2: Express the Perimeter Equation

The perimeter of a rectangle is given by the formula: $2(l + w) = 70$, where $l$ is the length and $w$ is the width. Simplify to $l + w = 35$.

3Step 3: Express the Diagonal Equation using Pythagorean Theorem

The diagonal of a rectangle can be found using the Pythagorean theorem: $d^2 = l^2 + w^2$. Here, $d = 25$, so $25^2 = l^2 + w^2$, resulting in $625 = l^2 + w^2$.

4Step 4: Solve the System of Equations

We now have two equations: 1) $l + w = 35$ and 2) $l^2 + w^2 = 625$. Solve this system by expressing $w$ from the first equation: $w = 35 - l$, and substitute in the second equation. This gives $l^2 + (35 - l)^2 = 625$.

5Step 5: Simplify and Solve the Quadratic Equation

Expand and simplify the equation: $l^2 + (35 - l)^2 = 625$ becomes $l^2 + 1225 - 70l + l^2 = 625$. This simplifies to $2l^2 - 70l + 600 = 0$. Divide the whole equation by 2: $l^2 - 35l + 300 = 0$.

6Step 6: Use the Quadratic Formula to Find Values of l

The quadratic formula is $l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Here, $a = 1, b = -35, c = 300$. Calculate the discriminant $b^2 - 4ac = (-35)^2 - 4(1)(300) = 1225 - 1200 = 25$. Hence, $l = \frac{35 \pm 5}{2}$. Thus, $l = 20$ or $l = 15$.

7Step 7: Find Corresponding Values for w

Substitute $l = 20$ into $w = 35 - l$, giving $w = 15$. Substitute $l = 15$, giving $w = 20$.

8Step 8: Verify Solutions

Check both solutions satisfy $l^2 + w^2 = 625$ since both sets $(l, w) = (20, 15)$ and $(l, w) = (15, 20)$ do satisfy this equation: - $20^2 + 15^2 = 400 + 225 = 625$ - $15^2 + 20^2 = 225 + 400 = 625$. Both are valid.

Key Concepts

perimeter of rectanglePythagorean theoremquadratic formula

perimeter of rectangle

The perimeter of a rectangle is a measurement that shows the total length around the rectangle. It's calculated by adding all the sides together. Since a rectangle has two pairs of equal sides, we can take the length, denoted by $ l $, and the width, denoted by $ w $, to calculate the perimeter with the formula

$ P = 2(l + w) $

In our problem, the perimeter is given as 70, so the equation becomes

$ 2(l + w) = 70 $

Simplifying this, we divide both sides by 2, resulting in

$ l + w = 35 $

This equation tells us that together, the length and the width must sum to 35. This forms one part of the system of equations used to solve for $ l $ and $ w $. Understanding how to find the perimeter is key to solving many geometric problems involving rectangles.

Pythagorean theorem

The Pythagorean theorem is an essential principle in geometry that relates the sides of a right triangle. In a rectangle, the diagonal creates two right triangles.

The theorem states that the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, it's expressed as

$ a^2 + b^2 = c^2 $

where $ a $ and $ b $ are the legs, and $ c $ is the hypotenuse.
For our rectangle problem, the diagonal $ d $ is the hypotenuse; therefore, we write

$ l^2 + w^2 = d^2 $

Given the diagonal $ d $ is 25, the equation becomes

$ l^2 + w^2 = 625 $

This equation is critical for solving the problem because it gives us another relationship between $ l $ and $ w $ allowing us to use both equations together to find the solution.

quadratic formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations, which are equations of the form $ ax^2 + bx + c = 0 $. These equations often appear when solving geometric problems after expressing variables and simplifying them.