The base of a triangle is 6 cm greater than twice the height.The area is 28 cm2.Find the height and the base.
Let
H=height in cm.
Base = 2H+6 cm
Area = (1/2)*H*base = 28
H(2H+6)=2*28=56
2H²+6H-56 = 0
H²+3H-28=0
Factorize
(H-4)(H+7)=0
H=4
or H=-7 (negative rejected)
18
Let's assume the height of the triangle is 'h' cm.
According to the given information, the base of the triangle is 6 cm greater than twice the height. Therefore, the base would be (2h + 6) cm.
The formula to find the area of a triangle is:
Area = (base * height) / 2
We are given the area as 28 cm², so we can substitute the values into the formula:
28 = (2h + 6) * h / 2
Simplifying the equation, we have:
28 = (2h² + 6h) / 2
Multiplying both sides of the equation by 2 to eliminate the fraction:
56 = 2h² + 6h
Now, rearrange the equation to form a quadratic equation:
2h² + 6h - 56 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. Let's use the quadratic formula:
h = (-b ± √(b² - 4ac)) / 2a
In this case, a = 2, b = 6, and c = -56. Let's substitute these values into the formula:
h = (-6 ± √(6² - 4 * 2 * -56)) / (2 * 2)
Simplifying further:
h = (-6 ± √(36 + 448)) / 4
h = (-6 ± √484) / 4
h = (-6 ± 22) / 4
This gives us two possible solutions for the height:
1) h = (-6 + 22) / 4 = 16 / 4 = 4 cm
2) h = (-6 - 22) / 4 = -28 / 4 = -7 cm
Since the height of a triangle cannot be negative, we discard the second solution.
So, the height of the triangle is 4 cm. Now, we can find the base:
Base = 2h + 6
Base = 2 * 4 + 6
Base = 8 + 6
Base = 14 cm
Therefore, the height of the triangle is 4 cm and the base is 14 cm.
To find the height and base of the triangle, you can use the formula for the area of a triangle:
Area = (base * height) / 2
Let's break down the information given in the problem:
1. The area of the triangle is 28 cm^2.
2. The base is 6 cm greater than twice the height.
Let's use algebraic equations to solve this problem step by step.
Let's assume the height of the triangle is "h" cm.
According to the information given, the base of the triangle is (2h + 6) cm.
Now, let's substitute the values into the formula for the area of a triangle:
28 = ((2h + 6) * h) / 2
Multiply both sides of the equation by 2 to eliminate the denominator:
56 = (2h + 6) * h
Expand the equation:
56 = 2h^2 + 6h
Rearrange the equation into a quadratic form:
2h^2 + 6h - 56 = 0
Now, we have a quadratic equation. We can solve it by factoring, using the quadratic formula, or completing the square method. In this case, let's use factoring.
Factor the quadratic equation:
(2h + 14)(h - 4) = 0
Now, set each factor equal to 0:
2h + 14 = 0 or h - 4 = 0
Solve for "h":
For 2h + 14 = 0, subtract 14 from both sides:
2h = -14
h = -7
For h - 4 = 0, add 4 to both sides:
h = 4
Since the height of a triangle cannot be negative, we discard the negative value. Thus, the height of the triangle is h = 4 cm.
Now, substitute the value of the height back into the equation for the base:
base = 2h + 6 = 2(4) + 6 = 8 + 6 = 14 cm
Therefore, the height of the triangle is 4 cm and the base is 14 cm.