Chapter 8.5, Problem 43PPE

a)

To determine

To find: Formula for the area of a parallelogram.

Answer to Problem 43PPE

Area of a parallelogram

  $=\text{ Height}\times \text{ base }$

Explanation of Solution

Given information: Any parallelogram is given.

Formula used: There is following prebuilt formula for the area of any parallelogram,

Area of a parallelogram $=\text{ Height}\times \text{ base }$ .

Calculation: In a parallelogram, there form two triangles with opposite equal bases and same heights. So,

  $\begin{array}{l}\text{Area of parallelogram}=2(\text{Area of any triangle)}\\ \text{=2}\times \frac{1}{2}\times \text{ height}\times \text{base }\\ \text{=}\text{Height}\times \text{base }\end{array}$

Conclusion: So, area of parallelogram is the product of its height and base.

b)

To determine

To find: If the constant term of height of parallelogram given is positive or negative.

Answer to Problem 43PPE

Height has a negative constant term.

Explanation of Solution

Given information: Area of parallelogram is $x^2-14x+24$ and its base is $(x-2)$ .

Formula used: Get two factors of quadratic polynomial, representing area of parallelogram and its second factor, except $(x-2)$ will be its height of parallelogram.

Calculation: Get two factors of given quadratic polynomial as,

  $\begin{array}{l}{x}^2-14x+24=x^2-(12+2)x+24\\ =x^2-12x-2x+24\\ =x(x-12)-2(x-12)\\ =(x-2)(x-12)\end{array}$

On comparing it with above formula of area of parallelogram, height of the parallelogram is $(x-12)$ .

Conclusion: By height obtained, it is cleared that constant term of height is negative.