I have tried multiple times on…

Maths Questions

Step by Step Solution

Mathematics Questions

I have tried multiple times on Question 15 and don’t understand it. Matikas is writing a coordinate proof to show that the midpoints of a quadrilateral form the vertices of a parallelogram. He assigns coordinates to the vertices of quadrilateral RSTV and labels the midpoints as A, B, C, and D. Can you provide the answers in simplified form to complete the proof?

Short Answer

The answer outlines a three-step process to determine coordinates of points A, B, C, and D and calculate the slopes of lines AD and BC. It uses the midpoint formula for coordinates and confirms that the slopes of both lines are equal at (a – c) / b.

Step-by-Step Solution

Step 1: Determine Coordinates of Points A and C

Using the midpoint formula, we can find the coordinates of points A and C. For point A, we take the midpoint of the line segment joining points R (0, 0) and S (2a, 2b), leading to coordinates (a, b). Similarly, for point C, which is the midpoint between points T (2c, 2d) and V (2c, 0), we get the coordinates (2c, d).

Step 2: Identify Coordinates of Points B and D

Next, we determine the coordinates of points B and D. The coordinates for point B are derived as (a+c, b+d), while point D has fixed coordinates (c, 0). This establishes the necessary points to proceed with slope calculations.

Step 3: Calculate the Slopes of Lines AD and BC

To find the slopes, we apply the slope formula, which is defined as slope = (y2 – y1) / (x2 – x1). For line AD, the slope is expressed as (c – a) / (0 – b), simplifying to (a – c) / b. Likewise, the computed slope of line BC results in the same expression, confirming that the slopes of lines AD and BC are indeed equal at (a – c) / b.

Related Concepts

Midpoint formula

A mathematical method used to find the exact center point of a line segment by averaging the x-coordinates and y-coordinates of the endpoints.

Slope formula

A mathematical expression used to determine the steepness or incline of a line, calculated as the difference in y-coordinates divided by the difference in x-coordinates of two points on the line.

Coordinates

A set of values that show an exact position on a two-dimensional plane, typically expressed as (x, y) pairs representing the horizontal and vertical positions.

Table Of Contents

I have tried multiple times on Question 15 and don't understand it. Matikas is writing a coordinate proof to show that the midpoints of a quadrilateral form the vertices of a parallelogram. He assigns coordinates to the vertices of quadrilateral RSTV and labels the midpoints as A, B, C, and D. Can you provide the answers in simplified form to complete the proof?