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1. Find the area of the convex quadrilateral with vertices (1, 5), (2, 3), (7, 6) and (7, 1).

 

 

2. A line goes through (2, 3) and (−8,−2). The line has an x-intercept of P and a y-intercept of Q. Let the origin be O. Find the area of triangleOPQ

 

3. ABCDEFGH is a rectangular prism with CD = 5, AD = 6, and AE = 8. Find the volume of pyramid ADCH..

 

 Jun 17, 2020
 #1
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For Q1, we can use the formula:

 

Area of triangle with vertices (x1, y1), (x2, y2), and (x3, y3) = \(\dfrac12 \cdot \left|\det\begin{bmatrix}1&x_1&y_1\\1&x_2&y_2\\1&x_3&y_3\end{bmatrix}\right|\)

 

Therefore, the area of the quadrilateral is \(\dfrac12 \cdot \left(\left|\det\begin{bmatrix}1&1&5\\1&2&3\\1&7&6\end{bmatrix}\right|+ \left|\det\begin{bmatrix}1&2&3\\1&7&6\\1&7&1\end{bmatrix}\right|\right) = \boxed{19\text{ squared units}}\)

 

I will leave the evaluation of determinants to you.

 Jun 17, 2020
 #2
avatar+8341 
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Q2)

 

Let P = (p, 0) and Q = (0, q).

 

Now, we find the equation of the straight line.

 

\(\text{Slope} = \dfrac{3-(-2)}{2-(-8)} = \dfrac12\)

 

Equation is \(y - 3 = \dfrac{x - 2}2\).

 

Now, you just have to find the values of p and q using the equation.

The area of triangle OPQ is |pq|/2 

 Jun 17, 2020
edited by MaxWong  Jun 17, 2020
 #3
avatar+8341 
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Q3)

Volume of pyramid ADCH = \(\dfrac16\left|\det\begin{bmatrix}0&0&6\\0&5&0\\8&0&0\end{bmatrix}\right| = 40\)

 Jun 17, 2020

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