Math  /  Geometry

Questionrest of the treasure will be under your feet.
10. Carefully draw all three medians on AGC\triangle A G C. Name the coordinates of yes theig

Label the midpoints as follows HH is the midpoint of AC\overline{A C} NN is the midpoint of EE is the midpoint
12. Determine the equation of median AE\overline{A E}. Show your work.

Studdy Solution

STEP 1

What is this asking? Find the equation of the line connecting point AA to the midpoint of the opposite side (GCGC) in the triangle AGCAGC. Watch out! Don't mix up the coordinates when calculating the midpoint and the slope!

STEP 2

1. Find Midpoint E
2. Calculate Slope of AE
3. Determine Equation of AE

STEP 3

Alright, so we need to find the midpoint EE of the line segment GCGC.
Remember, the midpoint formula is like finding the average of the x-coordinates and the average of the y-coordinates.
It's like meeting in the middle!

STEP 4

The coordinates of GG are (36,036, 0) and the coordinates of CC are (12,2412, 24).
Let's **plug these values** into our midpoint formula: E=(36+122,0+242) E = \left( \frac{36 + 12}{2}, \frac{0 + 24}{2} \right)

STEP 5

Now, let's **crunch those numbers**: E=(482,242)=(24,12) E = \left( \frac{48}{2}, \frac{24}{2} \right) = (24, 12) So, the coordinates of midpoint EE are (24,1224, 12).
Awesome!

STEP 6

Now, we need to find the slope of the line segment AEAE.
Remember, the slope is the "rise over run," or how much the y-value changes divided by how much the x-value changes.

STEP 7

We know the coordinates of AA are (0,00, 0) and the coordinates of EE are (24,1224, 12).
Let's **plug these values** into our slope formula: m=120240 m = \frac{12 - 0}{24 - 0}

STEP 8

Time to **simplify**: m=1224=12 m = \frac{12}{24} = \frac{1}{2} So, the **slope** of AEAE is 12\frac{1}{2}.
Perfect!

STEP 9

We've got our slope and a point on the line, so we can use the point-slope form of a linear equation, which is yy1=m(xx1)y - y_1 = m(x - x_1), where mm is the slope and (x1,y1x_1, y_1) is a point on the line.

STEP 10

We'll use point AA (0,00, 0) and our **slope** of 12\frac{1}{2}.
Let's **plug those in**: y0=12(x0) y - 0 = \frac{1}{2}(x - 0)

STEP 11

Now, let's **simplify**: y=12x y = \frac{1}{2}x Boom! There's our equation!

STEP 12

The equation of the median AE\overline{AE} is y=12xy = \frac{1}{2}x.

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