the base of a triangle is 3cm greater thanthe height.the area is 27cm find the heightand base of the triangle
height -- x
base -- x+3
(1/2)x(x+3) = 27
x^2 + 3x = 54
x^2 + 3x - 54 = 0
(x+9)(x-6) = 0
x = -9 (no good) or x = 6
height = 6
base = 9
To find the height and base of the triangle, let's use the formula for the area of a triangle:
Area = (1/2) * base * height
We know that the area is 27 cm², so we can substitute this value into the formula:
27 = (1/2) * base * height
Since the base is 3 cm greater than the height, we can express the base as (height + 3). Substituting this into the equation, we get:
27 = (1/2) * (height + 3) * height
To solve for the height, we can rearrange the equation:
54 = (height + 3) * height
Expanding the equation:
54 = height^2 + 3height
Rearranging to form a quadratic equation:
height^2 + 3height - 54 = 0
Now, we can solve this equation for the height using factoring, completing the square, or quadratic formula:
(height + 9)(height - 6) = 0
Setting each factor to zero:
height + 9 = 0 OR height - 6 = 0
Solving for height:
height = -9 OR height = 6
We can disregard the negative value since height cannot be negative in this context. Therefore, the height of the triangle is 6 cm.
To find the base, we can substitute the height value into the equation for the base:
base = height + 3 = 6 + 3 = 9 cm
Thus, the height of the triangle is 6 cm and the base is 9 cm.
To find the height and base of the triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Given that the area is 27 cm², we can substitute the values into the formula and solve for the unknown variables. Let's name the height as 'h' and the base as 'b'.
27 = (1/2) * b * h
Now, we are also given that the base is 3 cm greater than the height. We can express this as:
b = h + 3
Substituting this value for 'b' in the area formula:
27 = (1/2) * (h + 3) * h
Now, let's simplify the equation and solve for 'h':
54 = (h + 3) * h
Expanding the equation:
54 = h² + 3h
Rearranging the equation to make it quadratic:
h² + 3h - 54 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula:
(h + 9)(h - 6) = 0
From this equation, we find two possible values for 'h': h = -9 or h = 6. Since the height cannot be negative, we discard h = -9 and consider h = 6.
Now, substitute the value of 'h' back into the equation for 'b':
b = h + 3
b = 6 + 3
b = 9
So, the height of the triangle is 6 cm and the base is 9 cm.