Math / Geometry

QuestionWhat is the area of the triangle? $\square$ square units

Studdy Solution

STEP 1

What is this asking? We need to find the area of a triangle drawn on a graph. Watch out! Don't just assume it's a right triangle and use the legs as base and height!
Always double-check!

STEP 2

1. Find the base
2. Find the height
3. Calculate the area

STEP 3

Let's look at the triangle's vertices: $(-4, 4)$ , $(-4, -3)$ , and $(2, 2)$ .
Notice how two points have the same $x$ -coordinate, $-4$ .
This means they sit on a vertical line, which is super convenient!
This vertical line segment will be our base.

STEP 4

The two points are $(-4, 4)$ and $(-4, -3)$ .
To find the length of the base, we just find the difference in their $y$ -coordinates: $4 - (-3) = 4 + 3 = \mathbf{7}$ .
So, our base is $\mathbf{7}$ units long.

STEP 5

The height of the triangle is the perpendicular distance from the third vertex, $(2, 2)$ , to our base.
Since our base is vertical, the height will be a horizontal line segment.
This means we just need to find the horizontal distance between the vertex $(2, 2)$ and the line $x = -4$ .

STEP 6

The horizontal distance is the difference in the $x$ -coordinates: $2 - (-4) = 2 + 4 = \mathbf{6}$ .
So, our height is $\mathbf{6}$ units.

STEP 7

Now we have our base, $\mathbf{7}$ , and our height, $\mathbf{6}$ .
The formula for the area of a triangle is $\frac{1}{2} \cdot \text{base} \cdot \text{height}$ .

STEP 8

Let's plug in our values: $\text{Area} = \frac{1}{2} \cdot 7 \cdot 6$ $\text{Area} = \frac{1}{2} \cdot 42$ $\text{Area} = 21$

STEP 9

The area of the triangle is $\mathbf{21}$ square units.

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