What is the area of the parallelogram whose vertices are , and ?
Area Ex. 1
What is the area of a triangle whose vertices are , and ?
Area Ex. 1.2

Ask by Vaughn Lindsey. in the United States

Mar 20,2025

Question

What is the area of the parallelogram whose vertices are , and ?
Area Ex. 1
What is the area of a triangle whose vertices are , and ?
Area Ex. 1.2

Ask by Vaughn Lindsey. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

We first identify two vectors for the parallelogram using the vertex at $$ (0,0) $$.

1. Define vectors corresponding to two adjacent sides:
   - From $$ (0,0) $$ to $$ (-5,-6) $$: 
     $$
     \vec{u} = \langle -5, -6 \rangle
     $$
   - From $$ (0,0) $$ to $$ (7,-7) $$:
     $$
     \vec{v} = \langle 7, -7 \rangle
     $$

2. The area of the parallelogram is the magnitude of the cross product of these two vectors:
   $$
   \text{Area} = \left| \vec{u} \times \vec{v} \right| = \left| (-5)(-7) - (-6)(7) \right|
   $$
   Simplifying:
   $$
   (-5)(-7) = 35 \quad \text{and} \quad (-6)(7) = -42
   $$
   Therefore:
   $$
   \text{Area} = \left| 35 - (-42) \right| = \left| 35 + 42 \right| = 77
   $$

3. For the triangle whose vertices are $$ (0,0) $$, $$ (-5,-6) $$, and $$ (2,-13) $$, we use the formula for the area of a triangle given by three vertices:
   $$
   \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|
   $$
   Where:
   $$
   (x_1,y_1) = (0,0), \quad (x_2,y_2) = (-5,-6), \quad (x_3,y_3) = (2,-13)
   $$

4. Plug in the coordinates:
   $$
   \text{Area} = \frac{1}{2} \left| 0[(-6)-(-13)] + (-5)[(-13)-0] + 2[0-(-6)] \right|
   $$
   Simplify each term:
   $$
   0[7] = 0,\quad (-5)[-13] = 65,\quad 2[6] = 12
   $$
   Combining the terms:
   $$
   \text{Area} = \frac{1}{2} \left| 0 + 65 + 12 \right| = \frac{1}{2} \times 77 = \frac{77}{2}
   $$

Thus, the computed answers are:

- Parallelogram area: $$77$$
- Triangle area: $$\frac{77}{2}$$
The area of the parallelogram is 77, and the area of the triangle is 38.5.

What is the area of the parallelogram whose vertices are , and ?
Area Ex. 1
What is the area of a triangle whose vertices are , and ?
Area Ex. 1.2

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Geometry Questions

What is the area of the parallelogram whose vertices are , and ? Area Ex. 1 What is the area of a triangle whose vertices are , and ? Area Ex. 1.2

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Geometry Questions

What is the area of the parallelogram whose vertices are , and ?
Area Ex. 1
What is the area of a triangle whose vertices are , and ?
Area Ex. 1.2