Math Problem Statement

In the figure, ABCD is a rectangle, and P is a point on the side DC extended. Q is the midpoint of the line AP. The area of the triangle BPQ is 20 square centimeters. What is the area of the rectangle?

Solution

The problem you've shared involves a rectangle ABCDABCD and an extended point PP on the side DCDC. Let's break it down based on the details visible in the image:

  • ABCDABCD is a rectangle.
  • PP is a point on the side DCDC extended.
  • QQ is the midpoint of the line APAP.
  • The area of the triangle BPQBPQ is given as 20 square centimeters.
  • We are asked to find the area of the rectangle ABCDABCD.

Strategy:

  1. Set variables: Let AB=hAB = h (the height of the rectangle) and BC=wBC = w (the width of the rectangle). Therefore, the area of the rectangle is Arect=h×wA_{\text{rect}} = h \times w.

  2. Calculate the area of triangle BPQBPQ: The area of triangle BPQBPQ is given as 20 cm². The area of a triangle is calculated using the formula: A=12×base×heightA_{\triangle} = \frac{1}{2} \times \text{base} \times \text{height} Here, the base of the triangle BPBP would be along the extended side of the rectangle, and QQ, being the midpoint of APAP, adds complexity to finding its height.

  3. Relating triangle area to rectangle: By analyzing the geometric properties and given data, we need to use proportions or other geometric relations to relate the area of triangle BPQBPQ with the area of rectangle ABCDABCD.

I'll proceed with these calculations now.From the solution, we get that h=80wh = \frac{80}{w}. This indicates a relationship between the height hh and width ww of the rectangle based on the given conditions and the area of the triangle.

To calculate the area of the rectangle, we find: Area of the rectangle=h×w=80w×w=80\text{Area of the rectangle} = h \times w = \frac{80}{w} \times w = 80 Thus, the area of the rectangle is 80 square centimeters.

Would you like more details on the geometric reasoning or any clarification?

Here are 5 related questions:

  1. How would the problem change if point PP was inside the rectangle?
  2. What if triangle BPQBPQ's area was given as a different value?
  3. Can the area of triangle APQAPQ help verify the result?
  4. What geometric properties of midpoints help in problems like this?
  5. How does extending a side affect calculations in coordinate geometry?

Tip: Always check if a midpoint divides the figure symmetrically to simplify calculations.

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Math Problem Analysis

Mathematical Concepts

Geometry
Area of Triangles
Rectangles

Formulas

Area of a triangle = 1/2 × base × height
Area of a rectangle = height × width

Theorems

Midpoint Theorem

Suitable Grade Level

Grades 9-12