Question

Question #43f13

Answer 1

Rectangle

Explanation:

1. Find location of `B`:
   Given: The midpoint of `AB (1,5)` is up 3 and over 4 from `A`. `m_(AB) = 3/4`. This means `B` is up 3 and over 4 from (1,5): `B(5, 8), " and AB = 10`

2. Find location of `D`:
   Given: `m_(DA) = -4/3 = -1/(m_(AB))`, `" so " AB " perpendicular " DA = 90^@`
   From A go down 4, over 3: `D (0, -2)`
   Note: Since we don't know the length of `DA`, the location of D can vary.

3. Find location of `C`:
   Given: `CD` and `DA` are `90^@` this means they are perpendicular. `m_(DA) = -4/3 = -1/(m_(CD)), m_(CD) = 3/4 = m_(AB)`. This means `CD |\| AB`
   Given: `CD = 10`, using Pythagorean Theorem: `10^2 = (4x)^2 +(3x)^2`
   `100 = 16x^2 + 9x^2`
   `100 = 25x^2`
   `x^2 = 4`
   `x = 2`, `" so " 4x = 8`, `3x = 6`
   This means from `D` we need to go up 8, over 6 from `D` to find `C (8, 4)`
   Note: The location of `C` depends on `D`, which can vary. This means we don't know the length of `DA` or `AB`.

4. Find the `m_(BC ) = (y_2 - y_1)/(x_2 - x_1) = (8 - 4)/(5-8) = 4/-3 = -4/3` `" so " BC " is perpendicular to " AB`

So `ABCD = "rectangle"`