Questions: Unit 1: Section 1.5 Question 4 of 8 (1 point) Question Attempt: 1 of Unlimited The height of a triangle is 2 yd longer than the base x. The area is 84 yd². Part 1 of 3 (a) Write an equation in terms of x that represents the given relationship. The equation is 1/2 x(x+2)=84 Part: 1 / 3 Part 2 of 3 (b) Solve the equation to find the dimensions of the given shape. Be sure to include the correct unit symbol. The base is 12 yd

Solution Steps

To solve for the dimensions of the triangle, we need to solve the equation \(\frac{1}{2}x(x+2) = 84\) for \(x\). This equation represents the area of the triangle, where \(x\) is the base and \(x+2\) is the height. We will first multiply both sides by 2 to eliminate the fraction, then expand and rearrange the equation into a standard quadratic form. Finally, we will solve the quadratic equation using the quadratic formula or by factoring.

We start with the equation representing the area of the triangle, given by:

\[ \frac{1}{2}x(x + 2) = 84 \]

To eliminate the fraction, we multiply both sides by 2:

\[ x(x + 2) = 168 \]

Expanding this gives:

\[ x^2 + 2x - 168 = 0 \]

Using the quadratic formula, we find the roots of the equation \(x^2 + 2x - 168 = 0\). The solutions are:

\[ x = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 1 \cdot (-168)}}{2 \cdot 1} \]

Calculating the discriminant:

\[ \sqrt{4 + 672} = \sqrt{676} = 26 \]

Thus, the solutions are:

\[ x = \frac{-2 + 26}{2} = 12 \quad \text{and} \quad x = \frac{-2 - 26}{2} = -14 \]

Since dimensions cannot be negative, we take \(x = 12\).

The height of the triangle is given by:

\[ \text{Height} = x + 2 = 12 + 2 = 14 \]

Final Answer