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Math  /  Geometry

QuestionQuestion 11
The endpoints of JK are given. Find the coordinates of the midpoint when J(1,−3)J(1,-3) and K(7,5)\mathrm{K}(7,5). M=(M=( □\square □\square ) Question 12
The midpoint MM and one endpoint of Line ABA B are given. Find the coordinates of the other endpoint when M(−4,5)M(-4,5) and A(−1,−3)A(-1,-3). B=1B=1 □\square □\square Question 13
Find AMA M AM=A M= □\square

Studdy Solution

STEP 1

1. The midpoint formula for a line segment with endpoints J(x1,y1)J(x_1, y_1) and K(x2,y2)K(x_2, y_2) is given by (x1+x22,y1+y22)\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right).
2. To find the other endpoint B(x2,y2)B(x_2, y_2) given the midpoint M(xm,ym)M(x_m, y_m) and one endpoint A(x1,y1)A(x_1, y_1), we use the midpoint formula in reverse.
3. The distance formula between two points A(x1,y1)A(x_1, y_1) and M(x2,y2)M(x_2, y_2) is given by (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

STEP 2

1. Calculate the midpoint for the endpoints J(1,−3)J(1, -3) and K(7,5)K(7, 5).
2. Calculate the coordinates of endpoint BB given the midpoint M(−4,5)M(-4, 5) and endpoint A(−1,−3)A(-1, -3).
3. Calculate the distance AMAM given the coordinates A(−1,−3)A(-1, -3) and M(−4,5)M(-4, 5).

STEP 3

Calculate the x-coordinate of the midpoint.
xm=x1+x22=1+72=82=4 x_m = \frac{x_1 + x_2}{2} = \frac{1 + 7}{2} = \frac{8}{2} = 4

STEP 4

Calculate the y-coordinate of the midpoint.
ym=y1+y22=−3+52=22=1 y_m = \frac{y_1 + y_2}{2} = \frac{-3 + 5}{2} = \frac{2}{2} = 1

STEP 5

Combine the coordinates to find the midpoint MM.
M=(4,1) M = (4, 1)

STEP 6

Calculate the x-coordinate of endpoint BB. Use the reverse midpoint formula:
xm=x1+x22  ⟹  2xm=x1+x2  ⟹  x2=2xm−x1 x_m = \frac{x_1 + x_2}{2} \implies 2x_m = x_1 + x_2 \implies x_2 = 2x_m - x_1
x2=2(−4)−(−1)=−8+1=−7 x_2 = 2(-4) - (-1) = -8 + 1 = -7

STEP 7

Calculate the y-coordinate of endpoint BB. Use the reverse midpoint formula:
ym=y1+y22  ⟹  2ym=y1+y2  ⟹  y2=2ym−y1 y_m = \frac{y_1 + y_2}{2} \implies 2y_m = y_1 + y_2 \implies y_2 = 2y_m - y_1
y2=2(5)−(−3)=10+3=13 y_2 = 2(5) - (-3) = 10 + 3 = 13

STEP 8

Combine the coordinates to find the endpoint BB.
B=(−7,13) B = (-7, 13)

STEP 9

Calculate the distance AMAM using the distance formula:
AM=(x2−x1)2+(y2−y1)2 AM = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
AM=(−4−(−1))2+(5−(−3))2=(−4+1)2+(5+3)2 AM = \sqrt{(-4 - (-1))^2 + (5 - (-3))^2} = \sqrt{(-4 + 1)^2 + (5 + 3)^2}
AM=(−3)2+(8)2=9+64=73 AM = \sqrt{(-3)^2 + (8)^2} = \sqrt{9 + 64} = \sqrt{73}
Solution:
1. Midpoint M=(4,1)M = (4, 1).
2. Endpoint B=(−7,13)B = (-7, 13).
3. Distance AM=73AM = \sqrt{73}.

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