To solve this problem, we need to follow a structured approach using algebra and the formula for the area of a triangle.
### Step-by-step solution:
1. Identify the given information:
- The area of the triangle is 18 cm².
- The height (h) is 1 cm more than twice the base (b).
2. Express the relationship mathematically:
- Let the base of the triangle be [tex]\( b \)[/tex] cm.
- The height of the triangle will be [tex]\( 2b + 1 \)[/tex] cm.
3. Use the area formula of a triangle:
The area of a triangle is given by:
[tex]\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
Plugging in the known values:
[tex]\[
18 = \frac{1}{2} \times b \times (2b + 1)
\][/tex]
4. Remove the fraction by multiplying both sides by 2:
[tex]\[
36 = b \times (2b + 1)
\][/tex]
5. Form a quadratic equation:
[tex]\[
36 = 2b^2 + b
\][/tex]
Rearrange to standard quadratic form:
[tex]\[
2b^2 + b - 36 = 0
\][/tex]
6. Solve the quadratic equation:
The quadratic equation [tex]\( 2b^2 + b - 36 = 0 \)[/tex] can be solved using the quadratic formula:
[tex]\[
b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\][/tex]
Here, [tex]\( A = 2 \)[/tex], [tex]\( B = 1 \)[/tex], and [tex]\( C = -36 \)[/tex].
7. Calculate the discriminant:
[tex]\[
\text{Discriminant} = B^2 - 4AC = 1^2 - 4(2)(-36) = 1 + 288 = 289
\][/tex]
8. Find the roots:
[tex]\[
b = \frac{-1 \pm \sqrt{289}}{4}
\][/tex]
[tex]\[
b = \frac{-1 \pm 17}{4}
\][/tex]
This gives two solutions:
[tex]\[
b = \frac{16}{4} = 4 \quad \text{and} \quad b = \frac{-18}{4} = -4.5
\][/tex]
9. Determine the valid solution:
Since the base of a triangle cannot be negative, we discard [tex]\( b = -4.5 \)[/tex], leaving:
[tex]\[
b = 4
\][/tex]
10. Calculate the height:
Using the expression [tex]\( h = 2b + 1 \)[/tex]:
[tex]\[
h = 2(4) + 1 = 8 + 1 = 9
\][/tex]
### Final Answer:
The base of the triangle is 4 cm, and the height is 9 cm.
