In triangle ABC,AB=10,AC=7 and BC=8
How do we need to slide out along AB so that the area of the overlapping region(the shaded triangle A'BD) is half the area of triangle ABC?
To find out how much we need to slide out along side AB so that the area of the shaded triangle A'BD is half the area of triangle ABC, we can follow these steps:
Step 1: Calculate the area of triangle ABC using Heron's formula:
First, calculate the semi-perimeter of triangle ABC:
s = (AB + AC + BC) / 2
= (10 + 7 + 8) / 2
= 12.5
Now, calculate the area of triangle ABC:
Area of triangle ABC = √(s(s - AB)(s - AC)(s - BC))
= √(12.5(12.5 - 10)(12.5 - 7)(12.5 - 8))
= √(12.5 * 2.5 * 5.5 * 4.5)
≈ √(2,557.81)
≈ 50.57
Step 2: Calculate half the area of triangle ABC:
Half the area of triangle ABC = Area of triangle ABC / 2
= 50.57 / 2
≈ 25.28
Step 3: Set up an equation to find the distance we need to slide out (let's call it x). The idea is to calculate the area of triangle A'BD for various values of x and find the value of x for which the area is equal to 25.28.
Let's consider triangle A'BD separately. The base of this triangle is x, and the height is the same as the height of triangle ABC.
The area of triangle A'BD can be calculated using the formula:
Area of triangle A'BD = (1/2) * base * height
= (1/2) * x * (height of ABC)
Step 4: Set up the equation for area of triangle A'BD:
Area of triangle A'BD = (1/2) * x * (height of ABC)
We want the area of triangle A'BD to be equal to 25.28, so we can write:
25.28 = (1/2) * x * (height of ABC)
Step 5: Calculate the height of triangle ABC using Heron's formula:
The height of triangle ABC can be calculated as follows:
height of ABC = (2 * (area of ABC)) / (AB)
Substituting the values:
height of ABC = (2 * 50.57) / 10
= 101.14 / 10
= 10.11
Step 6: Substitute the values in the equation for area of triangle A'BD:
25.28 = (1/2) * x * 10.11
Step 7: Solve for x by multiplying both sides by 2 and then dividing by 10.11:
2 * 25.28 = x * 10.11
x = (2 * 25.28) / 10.11
x ≈ 4.99
So, we need to slide out along side AB by approximately 4.99 units in order for the area of the shaded triangle A'BD to be half the area of triangle ABC.
Since the height would not change, the base of the shaded triangle, which we can't see,
has to be 5 units to be half the original triangle.
remember : area of triangle = (1/2)(base)(height)