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Math  /  Geometry

QuestionWhat is the area of a triangle with vertices at (−4,1),(−7,5)(-4,1),(-7,5), and (0,1)(0,1) ?

Studdy Solution

STEP 1

1. We are given the coordinates of the vertices of a triangle.
2. The vertices are (−4,1)(-4,1), (−7,5)(-7,5), and (0,1)(0,1).
3. We need to find the area of the triangle using the formula for the area of a triangle given its vertices.

STEP 2

1. Write down the formula for the area of a triangle given its vertices.
2. Substitute the coordinates of the vertices into the formula.
3. Calculate the area using the formula.

STEP 3

The formula for the area of a triangle with vertices (x1,y1)(x_1, y_1), (x2,y2)(x_2, y_2), and (x3,y3)(x_3, y_3) is given by:
Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|

STEP 4

Substitute the given coordinates (−4,1)(-4,1), (−7,5)(-7,5), and (0,1)(0,1) into the formula:
Let (x1,y1)=(−4,1)(x_1, y_1) = (-4, 1), (x2,y2)=(−7,5)(x_2, y_2) = (-7, 5), (x3,y3)=(0,1)(x_3, y_3) = (0, 1).
Area=12∣(−4)(5−1)+(−7)(1−1)+(0)(1−5)∣\text{Area} = \frac{1}{2} \left| (-4)(5-1) + (-7)(1-1) + (0)(1-5) \right|

STEP 5

Calculate the area using the substituted values:
=12∣(−4)(4)+(−7)(0)+(0)(−4)∣= \frac{1}{2} \left| (-4)(4) + (-7)(0) + (0)(-4) \right| =12∣−16+0+0∣= \frac{1}{2} \left| -16 + 0 + 0 \right| =12×16= \frac{1}{2} \times 16 =8= 8
The area of the triangle is:
8 \boxed{8}

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