The base of is numerically 3 less than its area, and the height is numerically 6 less than its area. Find the area of the triangle.I am lost.....
Well, if I were lost in a triangle, I would probably ask for directions from the vertices. But let me help you with this problem!
Let's break it down step by step. We know that the base of the triangle is numerically 3 less than its area. So let's say the area of the triangle is A, then the base would be A - 3.
Similarly, we know that the height of the triangle is numerically 6 less than its area. So the height can be expressed as A - 6.
Now, the formula for the area of a triangle is: A = (1/2) * base * height. Plugging in what we know, we have:
A = (1/2) * (A - 3) * (A - 6)
To solve for A, we can simplify the equation:
2A = (A - 3) * (A - 6)
Expanding the right side:
2A = A^2 - 9A + 18
Moving everything to one side:
A^2 - 11A + 18 = 0
Using some clowny quadratic factoring (or you can use the quadratic formula), we find that A = 2 or A = 9.
Now, since we're talking about the area of a triangle, we can't have a negative area. So our solution is A = 9.
Therefore, the area of the triangle is 9 square units. Ta-da!
b = bh/2 - 3
h = bh/2 - 6
bh/2 = b+3 = h+6
so, h = b-3
b = b/2 (b-3) - 3
2b = b^2-3b-6
b^2-5b-6 = 0
b = 6 or -1
pick b=6
so, h=3
bh/2 = 6*3/2 = 9
To find the area of the triangle, we need to use the formula:
Area = (base * height) / 2
Let's solve this step-by-step:
Step 1: Let's assign variables to the base and height of the triangle.
Let's call the base of the triangle "b" and the height of the triangle "h".
Step 2: Given that the base of the triangle is numerically 3 less than its area,
we can express this as:
b = (area - 3)
Step 3: Also given that the height of the triangle is numerically 6 less than its area,
we can express this as:
h = (area - 6)
Step 4: Now, substitute the expressions for base and height into the area formula:
Area = (base * height) / 2
Area = [(area - 3) * (area - 6)] / 2
Step 5: Simplify the expression for the area:
Area = [(area^2) - 9area + 18] / 2
Step 6: Multiply both sides of the equation by 2 to eliminate the fraction:
2 * Area = (area^2) - 9area + 18
Step 7: Rearrange the equation to form a quadratic equation in standard form:
(area^2) - 9area + 18 - 2 * Area = 0
Step 8: Combine like terms:
(area^2) - (9 + 2) * area + 18 = 0
(area^2) - 11area + 18 = 0
Step 9: Factor the quadratic equation:
(area - 2)(area - 9) = 0
Step 10: Set each factor equal to zero and solve for the area:
area - 2 = 0 OR area - 9 = 0
If area - 2 = 0, then area = 2.
If area - 9 = 0, then area = 9.
Step 11: Check if the values of area satisfy the given conditions.
If area = 2, then the base would be (2 - 3) = -1, which is not possible.
Therefore, the only valid solution is area = 9.
Step 12: Calculate the base and height using the area value:
base = area - 3 = 9 - 3 = 6
height = area - 6 = 9 - 6 = 3
Step 13: Substitute the base and height values into the area formula to find the area:
Area = (base * height) / 2
Area = (6 * 3) / 2
Area = 18 / 2
Area = 9
Therefore, the area of the triangle is 9 square units.
To find the area of a triangle, we need to know the lengths of its base and height. In this case, we are given that the base is numerically 3 less than its area, and the height is numerically 6 less than its area.
Let's denote the area of the triangle as "A", the base as "b", and the height as "h".
We can set up two equations based on the given information:
1. The base is numerically 3 less than its area: b = A - 3
2. The height is numerically 6 less than its area: h = A - 6
Now we can solve for the values of "b" and "h" in terms of "A" by substituting the equations:
b = (A - 3)
h = (A - 6)
To find the area of the triangle, we use the formula for the area of a triangle: A = (1/2) * base * height.
Substituting the values of "b" and "h" into the formula:
A = (1/2) * (A - 3) * (A - 6)
We can simplify the equation by distributing and combining like terms:
2A = A^2 - 9A + 18
Rearranging the terms and setting the equation to zero:
A^2 - 11A + 18 = 0
Now we can solve this quadratic equation. Factoring or using the quadratic formula, we find:
A = 2 or A = 9
Since the area of a triangle cannot be negative, we discard the solution A = 2.
Therefore, the area of the triangle is A = 9.