△ABC has vertices at A(−1,1), B(−7,1), and C(−3,6).

What is the area of △ABC?
15 units ^2
34 units ^2
35 units ^2
17 units ^2

Formula is:

A = | [ Ax ( B y - Cy ) + Bx ( Cy - Ay ) + Cx ( A y - By ) ] / 2 |

where:

Ax and Ay are the x and y coordinates of the point A

Bx and By are the x and y coordinates of the point B

Cx and Cy are the x and y coordinates of the point C

Two vertical bars mean "absolute value".

In this case:

Ax = - 1 , Ay = 1

Bx = - 7 , By = 1

Cx = - 3, Cy = 6

A = | [ Ax ( B y - Cy ) + Bx ( Cy - Ay ) + Cx ( A y - By ) ] / 2 |

A = | [ ( - 1 ) ∙ ( 1 - 6 ) + ( - 7 ) ∙ ( 6 - 1 ) + ( - 3 ) ( 1 - 1 ) ] / 2 |

A = | [ ( - 1 ) ∙ ( - 5 ) + ( - 7 ) ∙ ( 5 ) + ( - 3 ) ∙ 0 ) ] / 2 |

A = | ( 5 - 35 ) / 2 |

A = | - 30 / 2 |

A = | - 15 |

A = 15 units²

Notice AB is a horizontal line since A and B have the same y value

So a quick count shows AB = 6
and C(-3,6) is clearly 5 units above it, so we have the base and the height
Area = (1/2)(6)(5) = 15

This was lucky, the method Bosnian used is a general method and works for any
3 points, it is just one of many ways to do this in a general way.

Well, calculating the area of a triangle isn't as easy as cracking a joke, but I'll do my best! To find the area of triangle ABC, we can use the formula:

Area = 1/2 * base * height

First, let's find the length of the base. We can use the distance formula to find the distance between points A and B:

√((-7 - (-1))^2 + (1 - 1)^2)
√((-6)^2 + (0)^2)
√(36 + 0)
√36
6

So, the length of the base AB is 6 units.

Next, let's find the height of the triangle. We can use the difference in y-coordinates between points A and C:

6 - 1
5

So, the height of the triangle is 5 units.

Now, let's calculate the area:

Area = 1/2 * base * height
Area = 1/2 * 6 * 5
Area = 1/2 * 30
Area = 15.

Therefore, the area of triangle ABC is 15 units^2.

And that's no joke!

To find the area of triangle ABC, we can use the formula for the area of a triangle given the coordinates of its vertices.

Let's label the coordinates of the vertices as follows:
A(-1, 1)
B(-7, 1)
C(-3, 6)

To find the base of the triangle, we can use the distance formula between points A and B:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-7 - (-1))^2 + (1 - 1)^2)
= sqrt((-6)^2 + 0^2)
= sqrt(36 + 0)
= sqrt(36)
= 6

Now, we need to find the height of the triangle. We can use the distance formula between point A and the line BC, which is perpendicular to AB.

The slope of the line BC can be found using the formula:
slope = (y2 - y1)/(x2 - x1)
= (6 - 1)/(-3 - (-7))
= 5/4

Since BC is perpendicular to AB, the slope of BC is the negative reciprocal of the slope of AB. So the slope of BC is -4/5.

Now, we have the slope of BC and a point on the line (A), so we can write the equation of line BC in point-slope form:
y - y1 = m(x - x1)
y - 1 = (-4/5)(x - (-3))
y - 1 = (-4/5)(x + 3)
y - 1 = (-4/5)x - 12/5

Now, to find the height, we need to find the y-coordinate of point C, where the line BC intersects the y-axis.
Setting x = 0 in the equation y - 1 = (-4/5)x - 12/5:
y - 1 = (-4/5)(0) - 12/5
y - 1 = 0 - 12/5
y - 1 = -12/5
y = -12/5 + 1
y = -12/5 + 5/5
y = -7/5

So, the height of the triangle is the vertical distance between point A and the line BC, which is 7/5.

Now, we can calculate the area of the triangle using the formula:
Area = (1/2) * base * height
= (1/2) * 6 * 7/5
= 3 * 7/5
= 21/5
= 4.2

Therefore, the area of triangle ABC is approximately 4.2 units squared.

To find the area of triangle ABC, we can use the formula for the area of a triangle given its coordinates.

Step 1: Find the lengths of the sides of the triangle.
To find the lengths of the sides AB, BC, and CA, we can use the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

For example, the length of side AB can be found as follows:
AB = √((-7 - (-1))^2 + (1 - 1)^2)
= √((-6)^2 + 0^2)
= √(36)
= 6

Similarly, you can find the lengths of sides BC and CA.

Step 2: Use Heron's formula to calculate the area.
Once we have the lengths of the sides, we can use Heron's formula to find the area of the triangle. Heron's formula states that the area (A) of a triangle with side lengths a, b, and c is given by:

A = √(s(s - a)(s - b)(s - c))

where s is the semi-perimeter of the triangle, calculated as:

s = (a + b + c)/2

Now, substitute the lengths of the sides into Heron's formula to find the area of triangle ABC:

s = (AB + BC + CA)/2
= (6 + ? + ?)/2
= (?)/2

A = √(? * (? - 6)(? - ?)(? - ?))

To determine the specific value of the area, we will need the lengths of sides BC and CA. Could you provide those?