Question

given points a(1,2), b(4,6), and c(7,2), which formula can be used to prove that triangle abc is isosceles?
a. distance formula for ab only
b. distance formula for all sides
c. midpoint formula
d. slope formula
what must be true for a quadrilateral to be considered a rhombus?
a. diagonals are perpendicular
b. all sides are of equal length
c. at least two sides are parallel
d. all sides are parallel
what is the slope of a line segment between points (1,2) and (1,6)?
a. 0
b. 1
c. undefined
d. 4

Explanation:

Step1: Recall isosceles - triangle property

An isosceles triangle has two sides of equal length. To check side - lengths, we use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for all sides.

Step2: Recall rhombus definition

A rhombus is a quadrilateral with all sides of equal length.

Step3: Recall slope - formula

The slope formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \((1,2)\) and \((1,6)\), \(x_1 = 1,x_2 = 1,y_1 = 2,y_2 = 6\). Then \(m=\frac{6 - 2}{1 - 1}=\frac{4}{0}\), which is undefined.

Answer:

1. B. Distance formula for all sides
2. B. All sides are of equal length
3. C. Undefined