Application of Quadratic Equations

The base of a triangle is longer than twice its corresponding height by 1 cm. The area of the triangle is 105 cm(square). Find the height of the triangle

height ---- x

base ---- 2x + 1

(1/2)(base)(height) = area
(1/2)(2x+1)(x) = 105
x(2x+1) = 210
2x^2 + x - 210 = 0
(x-10)(2x+21) = 0
x = 10 or x = -21/2 , but obviously x has to be positive, so

x = 10
The height is 10 cm

check:
height is 10
base is 21
area = (1/2)(21)(10) = 105
answer is correct

To solve this problem, we can use the formula for the area of a triangle:

Area of a triangle = (1/2) * base * height

Given information:
Area of the triangle = 105 cm²
Base of the triangle is longer than twice its height by 1 cm.

Let's start by assigning variables to the unknowns in the problem:
Let the height of the triangle be 'h' cm.
According to the problem, the base of the triangle is longer than 2h by 1 cm. So, the base can be represented as (2h + 1) cm.

Now, we can plug in these values into the area formula and solve for the height.

105 = (1/2) * (2h + 1) * h

To simplify the equation, we can start by distributing:
105 = (1/2) * (2h^2 + h)

Next, we can remove the fraction by multiplying both sides of the equation by 2:
210 = 2h^2 + h

Rearrange the equation in standard quadratic form:
2h^2 + h - 210 = 0

Now, we have a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = 1, and c = -210.

To find the values of h, we can use the quadratic formula:
h = [-b ± √(b^2 - 4ac)] / (2a)

Substituting the values of a, b, and c into the quadratic formula:
h = [-1 ± √(1^2 - 4 * 2 * -210)] / (2 * 2)

Simplifying further:
h = [-1 ± √(1 + 1680)] / 4
h = [-1 ± √1681] / 4
h = (-1 ± 41) / 4

Now, we have two possible values for h:
h₁ = (-1 + 41) / 4
h₁ = 40 / 4
h₁ = 10

h₂ = (-1 - 41) / 4
h₂ = -42 / 4
h₂ = -10.5

Since a negative height doesn't make sense in this context, we can discard the negative value.

Therefore, the height of the triangle is 10 cm.

To find the height of the triangle, we can use quadratic equations. Let's break down the problem into steps:

Step 1: Let's define the variables:
- Let's assume the base of the triangle is 'b' cm.
- Let's assume the height of the triangle is 'h' cm.

Step 2: Use the given information to write equations:
- The base of the triangle is longer than twice its corresponding height by 1 cm:
b = 2h + 1 (Equation 1)

- The area of the triangle is 105 cm²:
1/2 * base * height = 105
(1/2) * b * h = 105
bh = 210 (Equation 2)

Step 3: Substitute Equation 1 into Equation 2:
(2h + 1) * h = 210
2h² + h = 210
Rearrange:
2h² + h - 210 = 0

Step 4: Solve the quadratic equation:
You can solve this equation by factoring, completing the square, or using the quadratic formula. Let's solve it using factoring:

2h² + h - 210 = 0
(2h - 15)(h + 14) = 0

Setting each factor equal to zero:
2h - 15 = 0 or h + 14 = 0

Solve for 'h':
Case 1: 2h - 15 = 0
2h = 15
h = 15/2
h = 7.5 cm

Case 2: h + 14 = 0
h = -14
Since the height of a triangle cannot be negative, we discard this solution.

Step 5: Final Answer:
The height of the triangle is 7.5 cm.