3. xyz with vertices x(1,5) and z(1,1) has an area of 10 units? what are the coordinates of
the third vertex?
n

The coordinates of point Y are (-4 , 1) OR (6 , 1) OR (-4 , 5) OR (6 , 5)

Step-by-step explanation:

* Lets explain how to solve the problem

- Δ XYZ with vertices X = (1 , 5) and Z = (1 , 1)

- The area of the triangle XYZ is 10 unites²

- We need to find the third vertex y

∵ Point X is (1 , 5) and Point Z is (1 , 1)

- Points X and Z have the same x-coordinate

∵ The side XZ is a vertical side

- The length of any vertical segment is the difference between

  the y-coordinates of its endpoints

∵ The y-coordinates of point X and Z are 5 and 1

∴ The length of XZ = 5 - 1 = 4 units

∵ The area of any triangle = 1/2 × base × height

∵ XZ is a vertical side

∴ XZ is the height of the Δ XYZ

∵ The area of the triangle is 10 unites²

- Substitute these value in the rule of the area

∴ 10 = 1/2 × base × 4 ⇒ 1/2 × 4 = 2

∴ 10 = 2 base

- Divide both sides by 2

∴ base = 5 units

∵ The base is ZY or XY and both of them are horizontal segments

∴ ZY = 5 units OR XY = 5 units

- The length of any horizontal segment is the difference between

  the x-coordinates of its endpoints

# If the side is ZY

∵ The y-coordinates of the endpoints of the horizontal segment are

   equal

∵ point Z is (1 , 1)

∴ The y-coordinate of Y is 1

∵ The length of ZY = 5 units

- The length of any horizontal segment is the difference between

  the x-coordinates of its endpoints

∴ 5 = x - 1 ⇒ OR ⇒ 5 = 1 - x

∴ x = 6 OR x = -4

∴ The x-coordinates of point Y are 6 or -4

∴ The coordinates of point Y are (6 , 1) or (-4 , 1)

# If the side is XY

∵ The y-coordinates of the endpoints of the horizontal segment are

   equal

∵ point X is (1 , 5)

∴ The y-coordinate of Y is 5

∵ The length of XY = 5 units

- The length of any horizontal segment is the difference between

  the x-coordinates of its endpoints

∴ 5 = x - 1 ⇒ OR ⇒ 5 = 1 - x

∴ x = 6 OR x = -4

∴ The x-coordinates of point Y are 6 or -4

∴ The coordinates of point Y are (6 , 5) or (-4 , 5)

* The coordinates of point Y are (-4 , 1) OR (6 , 1) OR (-4 , 5) OR (6 , 5)