Math  /  Geometry

Question```latex \text{3. Use each vertex with its corresponding altitude's slope to graph the three altitudes of the triangle on the grid in Item 1.} \\ \text{4. After drawing the altitudes, Geoff is surprised to see that all three altitudes meet at one point. State the coordinates of the point of concurrency.} \\
\text{Al is not convinced that the altitudes of the triangle are concurrent and wants to use algebra to determine the coordinates of the points of intersection of the altitudes.} \\ \text{5. To use algebra to find the point where the altitudes meet, we need to know the equations of the altitudes.} \\ \text{a. State the coordinates of one point on the altitude from } A \text{ to } \overline{CG}. \\ \text{b. Write the equation of the altitude from } A \text{ to } \overline{CG}. \\ \text{c. Write the equation of the altitude from } C \text{ to } \overline{AG}. \\ \text{d. Write the equation of the altitude from } G \text{ to } \overline{AC}. \\
\text{Extracted text from attached image:} \\ \text{Al, Geoff, and Cal decide that their first priority is to find the treasure. They begin with the first clue from Leon's poem.} \\
\text{At the point on this map where two altitudes cross, In a hole in the ground some treasure was tossed} \\
\text{Since they do not know which two altitudes Leon meant, Geoff decides to place a grid over the map and draw all three altitudes to find the coordinates of any points of intersection.} \\
\text{Slope formula } a = m \frac{y_{1} - y_{1}}{20} A(0,0) \\ \text{1. Determine the slopes of the three sides of } \triangle AGC. \\ \text{e slopes} \\ \text{ocals.} \\ AC \\ \text{il line} \\ \text{ine is} \\ \qquad \\ \begin{array}{l} AC h = \frac{0-0}{36-0} = \frac{AG}{36} = 0 \\ \frac{1}{24-0} \\ \frac{24}{12-0} = \frac{24}{12} = 2 \end{array} \\ \text{2. Determine the slopes of the altitudes of the triangle.} \\ \text{a. What is the slope of the altitude from } A \text{ to } \overline{CG} \text{? Justify your answer.} \\
A \text{ must be perpendicular to the slope of } CG, \text{ so (1)} \\ \text{b. What is the slope of the altitude from } C \text{ to } \overline{AG} \text{? Justify your answer.} \\ \text{undifined} \\ \text{slope } = 0 \\ \text{c. What is the slope of the altitude from } G \text{ to } \overline{AC} \text{? Justify your answer.} \\ \frac{2}{-\frac{1}{2}} ```

Studdy Solution
By graphing, you should see the altitudes intersect at the point (12,12)(12,12).
Algebraically, we can find the intersection of y=xy=x and x=12x=12.
Substituting x=12x=12 into y=xy=x gives us y=12y=12.
Since the third line, y=0y=0, isn't involved in finding this intersection (because it intersects at a different point with each of the other lines), we confirm the intersection point is indeed (12,12)(12,12).

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