Question

The perimeter of a rectangle and an equilateral triangle are same. Also, one of the sides of the rectangle is equal to the side of the triangle. The ratio of the areas of the rectangle and the triangle is : (a) \(\sqrt{3}: 1\) (b) \(1: \sqrt{3}\) (c) \(2: \sqrt{3}\) (d) \(4: \sqrt{3}\)

Short answer

a) 1: 1 b) 1: 2 c) 2: √3 d) Cannot be determined

Step-by-step solution

Write the equation for the perimeters of both shapes

Let's write the equations for the perimeters of the rectangle and the equilateral triangle. Perimeter of the rectangle = 2(a + b) Perimeter of the equilateral triangle = 3a From the problem, we know that both perimeters are equal. Hence we can write 2 (a + b) = 3a.

Solve for b

Now, we will solve for b using the equation 2(a + b) = 3a. 2(a + b) = 3a 2a + 2b = 3a 2b = a b = a / 2

Calculate the areas for the rectangle and triangle

Now that we have the value of b, we can find the areas of the rectangle and the equilateral triangle. Area of Rectangle = a * b = a * (a / 2) = (1 / 2) * a^2 Area of Equilateral Triangle = \((\sqrt{3} / 4) * a^2\)

Calculate the ratio of the areas

Now, let's calculate the ratio of the areas of the rectangle and the triangle. \((Area of Rectangle) : (Area of Equilateral Triangle) = (\frac{1}{2}a^2):(\frac{\sqrt{3}}{4}a^2)\) To simplify the ratio, we need to divide both sides of the ratio by a^2. \((\frac{1}{2}):(\frac{\sqrt{3}}{4})\) Now, multiply both sides of the ratio by 4 to get rid of the fractional denominators \((\frac{1}{2} * 4) : (\frac{\sqrt{3}}{4} * 4) = 2 : \sqrt{3}\) So, the ratio of the areas of the rectangle and the triangle is \(2: \sqrt{3}\). The correct answer is (c) \(2: \sqrt{3}\).

Key concepts

Perimeter of Rectangle

The perimeter of a rectangle is the total distance around its outside edges. In simpler terms, if you take a walk around the rectangle, starting at one corner and making your way around until you get back to the starting point, the perimeter is the length of that walk. Mathematically, it's calculated by adding together the lengths of all four sides. Since opposite sides of a rectangle are equal in length, the formula to calculate the perimeter is:\[\begin{equation}P = 2(l + w)\text{, where}\begin{cases}P & \text{is the perimeter of the rectangle,}\l & \text{is the length of the rectangle,}\w & \text{is the width of the rectangle.}\text{
}\end{cases}\text{
}\end{equation}\]This concept is crucial, as you'll often need to find the perimeter when working with rectangles in geometric problems.

Perimeter of Equilateral Triangle

An equilateral triangle is a special triangle where all three sides are of equal length. To find the perimeter of an equilateral triangle, you simply add up the lengths of all three sides. Since they're all the same, it's even easier — you just multiply the length of one side by three.\[\begin{equation}P = 3s\text{, where}\begin{cases}P & \text{is the perimeter of the triangle, and}\s & \text{is the length of one side of the triangle.}\text{
}\end{cases}\text{
}\end{equation}\]Understanding perimeters is critical, especially when comparing the distances around different geometric shapes, as in the example exercise provided.

Area of Rectangle

The area of a rectangle represents the amount of space inside the four boundaries of the rectangle. To put it simply, if you were to paint the inside of a rectangle, the area would tell you how much paint you need. Calculated by multiplying the length by the width, the formula is:\[\begin{equation}A = l \times w\text{, where}\begin{cases}A & \text{is the area of the rectangle,}\l & \text{is the length, and}\w & \text{is the width.}\text{
}\end{cases}\text{
}\end{equation}\]This formula is one of the foundational concepts in geometry and is often used in various applications such as flooring, tiling, or mapping out space.

Area of Equilateral Triangle

The area of an equilateral triangle is slightly more complex to calculate than that of a rectangle because it involves a square root. For an equilateral triangle, the formula to find the area is:\[\begin{equation}A = \frac{\text{sqrt}{3} \times s^2}{4}\text{, where}\begin{cases}A & \text{is the area, and}\s & \text{is the length of one side.}\text{
}\end{cases}\text{
}\end{equation}\]To understand this, think of the formula as a way to measure the amount of space inside the triangle. The square root of three divided by four reflects the height's involvement in calculating an equilateral triangle's area.

Ratio of Areas

The ratio of areas compares the size of one area to another. It's like asking, how many times can one shape fit into another? For instance, if you calculate the area of two different shapes and find that one is twice as large as the other, the ratio of their areas is 2:1.In the given exercise, the ratio of the areas of a rectangle to an equilateral triangle is determined after calculating their respective areas with given side lengths. Remember, when working with ratios, ensure that any common factors in the numerator and denominator are simplified to get the most reduced form of the ratio.Understanding the concept of area helps with comparing spaces, say, between different lands, rooms, or any two-dimensional shapes. Ratios further allow one to quantify this comparison precisely.