a triangle is 6 cm wider that it is tall. The area is 56 cm². Find the height and the base.
W = T + 6
Area = 1/2 WT
Substitute T+6 for W in the second equation and solve for T. Insert that value into the first equation to solve for W. Check by putting both values into the second equation.
To solve this question, let's denote the height of the triangle as 'h' cm.
According to the problem, the triangle is 6 cm wider than it is tall. This means that the base of the triangle can be represented as 'h + 6' cm.
The formula to find the area of a triangle is:
Area = (1/2) * base * height
Given that the area of the triangle is 56 cm², we can substitute the values into the formula:
56 = (1/2) * (h + 6) * h
Next, we can simplify the equation by multiplying both sides by 2:
112 = (h + 6) * h
Now let's expand the equation:
112 = h^2 + 6h
Rearranging the equation into standard form:
h^2 + 6h - 112 = 0
Now we can solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, we will use factoring.
We find two numbers that multiply to -112 and add to 6. The numbers are 14 and -8:
(h + 14)(h - 8) = 0
Setting each factor equal to zero and solving for 'h':
h + 14 = 0 or h - 8 = 0
h = -14 or h = 8
Since the height of the triangle cannot be negative, we discard -14 as an extraneous solution.
Therefore, the height of the triangle is 8 cm.
To find the base, we can substitute the value of 'h' into the expression for the base:
Base = h + 6 = 8 + 6 = 14 cm
Therefore, the height of the triangle is 8 cm and the base is 14 cm.
To find the height and the base of the triangle, we can use the formula for the area of a triangle: Area = (1/2) * base * height.
We are given that the triangle is 6 cm wider than it is tall, so let's represent the height as 'h' cm. The base would then be 'h + 6' cm.
Substituting these values into the area formula: 56 cm² = (1/2) * (h + 6) * h.
To solve this equation, we need to isolate 'h'.
Let's start by multiplying both sides of the equation by 2 to get rid of the fraction: 2 * 56 cm² = (h + 6) * h.
Simplifying, we have: 112 cm² = h² + 6h.
Rearranging the equation, we get a quadratic equation: h² + 6h - 112 = 0.
To solve this quadratic equation, we can factor it, complete the square, or use the quadratic formula. Let's use the quadratic formula:
h = [-b ± √(b² - 4ac)] / (2a).
For our equation, a = 1, b = 6, and c = -112:
h = [-6 ± √((6)² - 4(1)(-112))] / (2(1)).
Simplifying further:
h = [-6 ± √(36 + 448)] / 2.
h = [-6 ± √484] / 2.
h = [-6 ± 22] / 2.
Now, we have two possible values for 'h'.
When h = (-6 + 22) / 2 = 16 / 2 = 8, the height is 8 cm.
When h = (-6 - 22) / 2 = -28 / 2 = -14, we discard this negative value since height cannot be negative in this context.
Therefore, the height of the triangle is 8 cm.
To find the base, we substitute this value back into the expression 'h + 6':
base = 8 + 6 = 14 cm.
So, the height of the triangle is 8 cm, and the base is 14 cm.