Find the missing side lengths, perimeter, and area of triangle ABC on the coordinate plane. All answers should be in simplest radical form, if necessary:
Side length AB:
Side length BC:
Side length AC:
Perimeter of triangle ABC:
Area of triangle ABC:
a is: A = (4,5)
b is: B = (1,2)
c is: C = (6,2)
Note: the formula for area of a triangle is A = bh/2, where b is the length of the base (BC) and h is the height from side BC to point A.
Answer choices:
1. √18
2. 9√2
3. 3√2
4. √4 + √9
5. √13
6. √5
7. 5
8. 5 + √13
9. 5 + 3√2 + √13
10. 15√26
11. 7.5
12. 15
To find the lengths of the sides of triangle ABC, we can use the distance formula. Let's calculate each side:
AB:
The coordinates of A and B are (4, 5) and (1, 2) respectively.
Using the distance formula, the length of AB is sqrt((4 - 1)^2 + (5 - 2)^2) = sqrt(9 + 9) = sqrt(18) = √18.
BC:
The coordinates of B and C are (1, 2) and (6, 2) respectively.
Using the distance formula, the length of BC is sqrt((1 - 6)^2 + (2 - 2)^2) = sqrt(25 + 0) = sqrt(25) = 5.
AC:
The coordinates of A and C are (4, 5) and (6, 2) respectively.
Using the distance formula, the length of AC is sqrt((4 - 6)^2 + (5 - 2)^2) = sqrt(4 + 9) = sqrt(13).
The lengths of the sides are:
AB = √18
BC = 5
AC = √13
To find the perimeter, we add the lengths of the sides:
Perimeter = AB + BC + AC = √18 + 5 + √13.
To find the area, we need to calculate the height from side BC to point A. We can find the slope of BC (since BC is parallel to the x-axis, the slope is 0), and using the point-slope form, we can write the equation of the line passing through point A and parallel to BC as y - 5 = 0(x - 4). Simplifying the equation gives us y = 5.
So, the height from side BC to point A is 5 units.
Now, we can use the area formula:
Area = (BC * height) / 2 = (5 * 5) / 2 = 25 / 2.
The missing values are:
Side length AB: √18
Side length BC: 5
Side length AC: √13
Perimeter of triangle ABC: √18 + 5 + √13
Area of triangle ABC: 25/2.
Therefore, the correct choices are:
AB: √18 (Option 1)
BC: 5 (Option 7)
AC: √13 (Option 5)
Perimeter: √18 + 5 + √13 (Option 8)
Area: 25/2 (Option 11).