Sharon made a scale drawing of a triangular park. The coordinates for the vertices of
the park are:
(– 10, 5)
(15, 5)
(10, 12)
Her scale is 1 unit = 1 meter.
What is the area of the triangular park in square meters?
How do you find the answer?
A quick sketch shows that your base is 25 units and the height is 7 units
area = (1/2)(25)(7) = 87.5 units^2
or 87.5 m^2
how did you know the base was 25 units and the height was 7 units?
To find the area of the triangular park, we can use the formula for the area of a triangle: A = (1/2) * base * height.
To calculate the base and height, we first need to find the lengths of the sides of the triangle using the given coordinates.
Let's label the vertices of the triangle as follows:
A = (-10, 5)
B = (15, 5)
C = (10, 12)
To find the length of AB, we can use the formula for distance between two points:
AB = √[(x2 - x1)² + (y2 - y1)²]
Plugging in the coordinates:
AB = √[(15 - (-10))² + (5 - 5)²]
= √[25² + 0²]
= √[625 + 0]
= √625
= 25
Similarly, we can find the lengths of BC and AC:
BC = √[(10 - 15)² + (12 - 5)²]
= √[(-5)² + 7²]
= √[25 + 49]
= √74
AC = √[(-10 - 10)² + (5 - 12)²]
= √[(-20)² + (-7)²]
= √[400 + 49]
= √449
Now that we have the lengths of all three sides (AB, BC, AC), we can find the area using Heron's formula, which states that the area of a triangle can be calculated using the lengths of its sides:
A = √[s(s-AB)(s-BC)(s-AC)]
where s is the semi-perimeter, given by:
s = (AB + BC + AC) / 2
Plugging in the values:
s = (25 + √74 + √449) / 2
Now we can calculate the area:
A = √[s(s-AB)(s-BC)(s-AC)]
By plugging in the values for s, AB, BC, and AC, we can find the area of the triangular park in square meters.