Problem 81
Question
The height of a triangle is \(6 \mathrm{~cm}\) less than the base. a. If \(b=\) base, write a polynomial expression in \(b\) that represents the height, and draw a diagram of the triangle. Do not include the units. b. Write a polynomial expression in \(b\) that represents the area.
Step-by-Step Solution
Verified Answer
Height: b-6; Area: \( \frac{1}{2}b^2 - 3b \)
1Step 1: Express the height in terms of the base
Given that the height of the triangle is 6 cm less than the base, if the base is represented as b, the height can be expressed as:height = b - 6.
2Step 2: Drawing the triangle
Visualize the triangle with the base labeled as b and the height labeled as b - 6. This helps in understanding the relationship between the base and the height.
3Step 3: Area formula of a triangle
The area of a triangle is given by the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
4Step 4: Substitute the expressions
Substitute the base (b) and the height (b - 6) into the area formula:\[ \text{Area} = \frac{1}{2} \times b \times (b - 6) \]
5Step 5: Simplify the polynomial expression for the area
Distribute and simplify the expression for the area:\[ \text{Area} = \frac{1}{2} \times (b^2 - 6b) = \frac{1}{2}b^2 - 3b \]
Key Concepts
Triangle Area Formula in AlgebraUnderstanding Algebraic ExpressionsSimplification of Polynomials
Triangle Area Formula in Algebra
The area of a triangle can be easily found using the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
This formula is derived from the basic geometric properties of a triangle.
It requires you to know both the base and the height of the triangle.
In our exercise, if the base is given as \(b\) and the height is \(b - 6\) (where height is 6 cm less than the base), we can calculate the area by substituting these expressions into the formula.
First, substitute the base: \(b\) and height: \(b - 6\) into the formula:
\[ \text{Area} = \frac{1}{2} \times b \times (b - 6) \]
By simplifying this expression, we get the final area formula for this specific triangle.
This formula is derived from the basic geometric properties of a triangle.
It requires you to know both the base and the height of the triangle.
In our exercise, if the base is given as \(b\) and the height is \(b - 6\) (where height is 6 cm less than the base), we can calculate the area by substituting these expressions into the formula.
First, substitute the base: \(b\) and height: \(b - 6\) into the formula:
\[ \text{Area} = \frac{1}{2} \times b \times (b - 6) \]
By simplifying this expression, we get the final area formula for this specific triangle.
Understanding Algebraic Expressions
Algebraic expressions are mathematical phrases that include numbers, variables, and operators (like add, subtract, multiply, and divide).
They are used to represent real-world problems in algebra.
In our example, the height of the triangle is given as 6 cm less than its base.
We represent the base as \(b\). So, the height can be expressed algebraically as \(b - 6\).
This simplification makes it easier to understand and solve the problem.
It also allows us to express the area of the triangle, another algebraic expression: \(\frac{1}{2} b^2 - 3b\).
Understanding and simplifying such expressions is key in algebra.
They are used to represent real-world problems in algebra.
In our example, the height of the triangle is given as 6 cm less than its base.
We represent the base as \(b\). So, the height can be expressed algebraically as \(b - 6\).
- \(b\) is the base.
- \(b - 6\) is the height.
This simplification makes it easier to understand and solve the problem.
It also allows us to express the area of the triangle, another algebraic expression: \(\frac{1}{2} b^2 - 3b\).
Understanding and simplifying such expressions is key in algebra.
Simplification of Polynomials
Simplification of polynomials involves combining like terms to make the expression easier to understand and work with.
In our problem, after substituting the base and height into the area formula, we get the expression:
\[ \text{Area} = \frac{1}{2} \times b \times (b - 6) \]
By distributing \(b\), this expression becomes:
\[ \text{Area} = \frac{1}{2} \times (b^2 - 6b) \]
We then distribute the \(\frac{1}{2}\) to each term within the parentheses:
\[ \text{Area} = \frac{1}{2}b^2 - 3b \]
Thus, we've simplified the polynomial expression for the area of the triangle.
It started as a polynomial multiplication and simplified into a combined form.
This final expression \(\frac{1}{2}b^2 - 3b\) enables easier calculations and a clearer understanding of the triangle's area in terms of its base.
In our problem, after substituting the base and height into the area formula, we get the expression:
\[ \text{Area} = \frac{1}{2} \times b \times (b - 6) \]
By distributing \(b\), this expression becomes:
\[ \text{Area} = \frac{1}{2} \times (b^2 - 6b) \]
We then distribute the \(\frac{1}{2}\) to each term within the parentheses:
\[ \text{Area} = \frac{1}{2}b^2 - 3b \]
Thus, we've simplified the polynomial expression for the area of the triangle.
It started as a polynomial multiplication and simplified into a combined form.
This final expression \(\frac{1}{2}b^2 - 3b\) enables easier calculations and a clearer understanding of the triangle's area in terms of its base.
Other exercises in this chapter
Problem 80
\(\left(\frac{3 a}{5 b}\right)^{2}(2 a)^{2}\)
View solution Problem 81
Problem: Use long division to simplify $$ \left(2 x^{2}+5 x+3\right) \div(x-8) $$ Incorrect Answer: $$ \begin{array}{r} x - 8 \longdiv { 2 x ^ { 2 } + 5 x + 3 }
View solution Problem 81
\(x\left(\frac{5}{6} x-8\right)+2\left(\frac{3}{4} x-9\right)\)
View solution Problem 81
\(C=\frac{7.7 \times 10^{-13}}{5 \times 10^{-6}}\)
View solution