The height of triangle is 4-cm more than twice the length of the base. The area of the triangle is 35 square centimeter. Find the height of the triangle.

Let's denote the base of the triangle as \( b \) cm and the height as \( h \) cm. According to the problem, the height is 4 cm more than twice the base:

\[
h = 2b + 4
\]

The formula for the area of a triangle is given by:

\[
\text{Area} = \frac{1}{2} \times b \times h
\]

We know the area is 35 square centimeters, so we can substitute the values:

\[
35 = \frac{1}{2} \times b \times (2b + 4)
\]

Simplify the equation:

\[
35 = \frac{1}{2} \times (2b^2 + 4b)
\]

\[
35 = b^2 + 2b
\]

Rearrange the equation into standard quadratic form:

\[
b^2 + 2b - 35 = 0
\]

Now, solve this quadratic equation using the quadratic formula:

\[
b = \frac{-b_1 \pm \sqrt{b_1^2 - 4ac}}{2a}
\]

Let’s plug in the values (\( a = 1, b = 2, c = -35 \)).

\[
b = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot -35}}{2 \cdot 1}
\]The solutions for the base are \( b = 5 \) cm and \( b = -7 \) cm. Since the base of a triangle cannot be negative, we take \( b = 5 \) cm.

Now, substitute the value of \( b \) into the equation for the height:

\[
h = 2b + 4 = 2(5) + 4 = 10 + 4 = 14 \, \text{cm}
\]

So, the height of the triangle is **14 cm**.

---

Would you like more details or have any questions? Here are some related questions:

1. How would the problem change if the area was different?
2. What if the height was a different multiple of the base?
3. How can we find the base if the height is known?
4. What are other ways to solve this type of problem?
5. How does changing the base affect the area?

**Tip:** When working with geometric problems, always double-check the units and relationships between variables.

Learn how to find the height of a triangle where the height is 4 cm more than twice the base and the area is 35 square centimeters. This step-by-step solution involves solving a quadratic equation and using the area of a triangle formula. Suitable for students in Grades 8-10.

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