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Chapter 7: Problem 65

The perimeter of a triangle is 180 feet. The longest side of the triangle is 9 feet shorter than twice the shortest side. The sum of the lengths of the two shorter sides is 30 feet more than the length of the longest side. Find the lengths of the sides of the triangle.

Short Answer

Expert verified
The lengths of the sides of the triangle, starting from the shortest, are 42 feet, 63 feet, and 75 feet.

Step by step solution

01

Translate the problem into algebraic equations

Let's call the shortest side of the triangle \( x \). The longest side, then, is \( 2x - 9 \). The other side, also shorter, is the remaining length that, added to the longest and the shortest side, totals the perimeter of the triangle, thus it is \( 180 - x - (2x - 9) = 189 - 3x \). The sum of the two shortest sides is equal to 30 feet more than the length of the longest side so we can write the equation \( x + (189-3x) = (2x - 9) + 30 \).
02

Solve the equations

First, simplify the equation to \( 189 - 2x = 2x + 21 \). Solving for \( x \), we get \( x = 42 \) feet. This gives us the length of the shortest side. Substitute \( x = 42 \) into \( 2x - 9 \) to find the longest side, which is \( 2(42) - 9 = 75 \) feet. The remaining side can be found by subtracting the known sides from the total perimeter, which will be \( 180 - 42 - 75 = 63 \) feet.
03

Verify that the solution is correct

The final step is to ensure that the solution set meets all the conditions specified in the problem, which states additionally that the sum of the lengths of the two shortest sides is 30 feet more than the length of the longest side. Adding the lengths of the two shortest sides (42 and 63 feet) gives us 105 feet, and, indeed, the length of the longest side (75 feet) plus 30 is also equal to 105 feet, so the conditions are met and the solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Equations
Algebraic equations are powerful tools that help us model and solve real-world problems. In this exercise, we need to find the lengths of the sides of a triangle given a set of conditions. We translate these conditions into algebraic expressions.
  • Assign variables to unknown quantities, such as the side lengths of the triangle.
  • Express each condition given in terms of these variables.

For example, we call the shortest side of the triangle \( x \). Then, based on the problem, the longest side is defined as \( 2x - 9 \). Another condition states that the sum of the two shorter sides is 30 feet more than the longest side. These expressions are key to forming equations that reveal the relationship between the sides.
Solving Equations
Once algebraic equations are established, solving them involves isolating the variables to find their values. The main goal is to simplify and manipulate the equations until the solution becomes clear.
  • Simplify complex expressions by combining like terms.
  • Use inverse operations to isolate the variable.

In our exercise, simplifying the equation results in \( 189 - 2x = 2x + 21 \). Moving terms involving \( x \) to one side and constants to the other helps isolate \( x \), leading to \( x = 42 \). This represents the shortest side length of the triangle.
Triangle Sides
Understanding the properties of triangle sides is crucial in geometry. Every triangle has three sides, and their lengths determine the shape or type of the triangle. The perimeter, the sum of all sides, limits the possible values each side can take.
  • In our problem, we have three sides with specific conditions about their lengths.
  • The given perimeter and relationships between the sides allow us to solve for each side.

After calculating \( x \), the shortest side, we use the expressions we formulated to find each specific side: the longest side is \( 75 \) feet, and the other shorter side is \( 63 \) feet.
Geometry Problem
Geometry problems often involve using known properties and relationships to find unknown measurements. In this triangle problem, we use basic geometric knowledge about perimeter and the properties of triangles.
  • The perimeter condition is used to check that all derived side measures fit within the bounds of the problem.
  • Verification ensures that all calculated side lengths adhere to the problem's conditions.

For instance, summing the two shorter side lengths \( 42 \) and \( 63 \) results in \( 105 \) feet, validating our calculations because it matches the longest side plus \( 30 \), as given. Solving geometry problems involves not just finding numbers but ensuring consistency with the given data and properties.

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Most popular questions from this chapter

An investor has up to \(450,000\) to invest in two types of investments. Type \(A\) pays \(6 \%\) annually and type B pays \(10 \%\) annually. To have a well- balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at least one-fourth of the portfolio is to be allocated to type \(B\) investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return?

Solve the system of linear equations and check any solutions algebraically. $$\left\\{\begin{aligned} x \quad\quad\quad\quad+3 w=4 \\ 2 y-z-w =0 \\\ \quad3 y \quad\quad-2 w=1 \\ 2 x-y+4 z\quad\quad =5 \end{aligned}\right.$$

Find the minimum and maximum values of the objective function and where they occur, subject to the constraints \(x \geq 0, y \geq 0, x+4 y \leq 20\) \(x+y \leq 18,\) and \(2 x+2 y \leq 21\). $$z=4 x+5 y$$

System A system of pulleys is loaded with 128-pound and 32 -pound weights (see figure). The tensions \(t_{1}\) and \(t_{2}\) in the ropes and the acceleration \(a\) of the 32-pound weight are found by solving the system of equations $$ \left\\{\begin{aligned} t_{1}-2 t_{2}\quad\quad =0 \\ t_{1}\quad\quad -2 a=128 \\ \quad t_{2}+a =32 \end{aligned}\right. $$ where \(t_{1}\) and \(t_{2}\) are in pounds and \(a\) is in feet per second squared. Solve this system.

Use a graphing utility to graph the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. Objective function: \(z=3 x+y\) Constraints: $$\begin{aligned}x & \geq 0 \\\y & \geq 0 \\ x+4 y & \leq 60 \\\3 x+2 y & \geq 48\end{aligned}$$
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