Pregunta
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  1. The mid points of sides and of triangle are and respectively. Find the length of side QR.
Find the area of the triangle formed by the points, and

Ask by Barker Curry. in India
Jan 30,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The length of side is 20 units, and the area of the triangle formed by points , , and is 47 square units.

Solución

Let’s solve each part of the problem step-by-step.

1. Finding the Length of Side in Triangle

Given:
  • Midpoint of is .
  • Midpoint of is .
Steps:
  1. Let the Coordinates of Point be .
  2. Find Coordinates of Points and :
    • Since is the midpoint of :
    • Since is the midpoint of :
  3. Determine the Distance :
    • Coordinates of :
    • Coordinates of :
    • Distance formula: Substituting the coordinates:
Answer:
The length of side is 20 units.

2. Finding the Area of Triangle Formed by Points , , and

Using the Shoelace Formula:
Substituting the Given Points:
Answer:
The area of the triangle is 47 square units.

Summary:
  • Length of side : 20 units
  • Area of triangle : 47 square units

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Bonus Knowledge

To find the length of side in triangle , we utilize the midpoint formula. The coordinates of midpoints and give us the following equations based on the coordinates of points :
We can solve these equations to find the coordinates of points . Since can be expressed as the straight-line distance between points and , we need to essentially derive these points (though the actual values are needed for the exact length).
However, let’s say we computed and ; the length can ultimately be found using the distance formula, .
For the area calculation of triangle with points , , and , we can use the formula:
Plugging in the values:
This simplifies to:
Calculating results in an area of 53. Thus, the side length can be found with the earlier established coordinates and the area of triangle is 53.

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