The Base of a Triangle: Solving for Height and Area with Quadratic Equations

In this article, we explore how to find the base of a triangle when given its height in relation to its base, and the area of the triangle. We use the formula for the area of a triangle along with quadratic equations to solve for the base. Let's go through each example step by step.

Scenario 1: Height of a Triangle is 5 cm More than Twice the Base

Given: The height of a triangle is 5 cm more than twice its base, and its area is 74 square cm. We need to find the base of the triangle.

1. Formula for the area of a triangle:
[text{Area} frac{1}{2} times text{base} times text{height}]

2. Let the base be (b) cm. Then, the height (h 2b 5) cm.

3. Using the area formula and substituting the values:

[frac{1}{2} times b times (2b 5) 74]

4. To eliminate the fraction, multiply both sides by 2:

[b times (2b 5) 148]

5. Expanding and rearranging the equation to form a quadratic equation:

[2b^2 5b - 148 0]

6. Using the quadratic formula:

[b frac{-B pm sqrt{B^2 - 4AC}}{2A}]

Where (A 2), (B 5), and (C -148).

7. Calculate the discriminant:

[B^2 - 4AC 5^2 - 4 times 2 times (-148) 25 1184 1209]

8. Substitute back into the quadratic formula:

[b frac{-5 pm sqrt{1209}}{4}]

9. Approximate (sqrt{1209} approx 34.8):

[b frac{-5 pm 34.8}{4}]

10. Calculate the two possible values for (b):

[b frac{-5 34.8}{4} approx 7.45] [b frac{-5 - 34.8}{4} frac{-39.8}{4} approx -9.95 text{ (not valid since base cannot be negative)}]

11. Thus, the base of the triangle is approximately 7.45 cm.

Scenario 2: Height is 3 cm More than the Base

Given: The height of a triangle is 3 cm more than the base, and its area is 14 square cm. We need to find the base of the triangle.

1. Using the area formula:

[frac{1}{2} times text{base} times text{height} 14]

2. Let the base be (x) cm. Then, the height (h x 3) cm.

3. Substitute the values into the area formula:

[frac{1}{2} times x times (x 3) 14]

4. Simplify and rearrange to form a quadratic equation:

[x^2 3x 28]

[x^2 3x - 28 0]

5. Factor the quadratic equation:

[x^2 7x - 4x - 28 0]

[x(x 7) - 4(x 7) 0]

[x - 4 0 text{ or } x 7 0]

[x 4 text{ or } x -7 text{ (not valid since base cannot be negative)}]

6. Thus, the length of the base of the triangle is 4 cm.

Scenario 3: Height is 3 cm More than the Length of the Base

Given: The height of a triangle is 3 cm more than the length of its base, and its area is 14 square cm. We need to find the base of the triangle.

1. Using the area formula:

[frac{1}{2} times text{base} times text{height} 14]

2. Let the base be (x) cm. Then, the height (h x 3) cm.

3. Substitute the values into the area formula:

[frac{1}{2} times x times (x 3) 14]

4. Simplify and rearrange to form a quadratic equation:

[x^2 3x - 28 0]

5. Use the quadratic formula:

[x frac{-3 pm sqrt{3^2 - 4 cdot 1 cdot (-28)}}{2 cdot 1}]

[x frac{-3 pm sqrt{9 112}}{2}]

[x frac{-3 pm sqrt{121}}{2}]

[x frac{-3 pm 11}{2}]

[x 4 text{ or } x -7 text{ (not valid since base cannot be negative)}]

6. Thus, the length of the base of the triangle is 4 cm.

Scenario 4: Height is 3 cm More than the Base

Given: The height of a triangle is 3 cm more than the base, and its area is 14 square cm. We need to find the base of the triangle.

1. Using the area formula:

[frac{1}{2} times text{base} times text{height} 14]

2. Let the base be (x) cm. Then, the height (h x 3) cm.

3. Substitute the values into the area formula:

[frac{1}{2} times x times (x 3) 14]

4. Simplify and rearrange to form a quadratic equation:

[x^2 3x 28]

[x^2 3x - 28 0]

5. Solve the quadratic equation:

[x frac{-3 pm sqrt{9 112}}{2}]

[x frac{-3 pm 11}{2}]

[x 4 text{ or } x -7 text{ (not valid since base cannot be negative)}]

6. Thus, the length of the base of the triangle is 4 cm.

Conclusion

In each scenario, we use the area formula and quadratic equations to find the base of the triangle. By solving these equations, we can accurately determine the base length needed for the triangle to achieve a specific area. These problems demonstrate the importance of understanding the relationship between the base, height, and area in geometric shapes.

Keywords: triangle area, base of triangle, quadratic equation