Math Problem Statement
The area of trapezoid ABCDABCD (AB∣∣CDAB∣∣CD) is 140 cm2.140 cm2. Through vertex DD we have constructed segment DNDN (with NN on side ABAB) such that DN∣∣BC.DN∣∣BC. If AB=20 cmAB=20 cm and CD=8 cmCD=8 cm, find the area of △AND.△AND.
Solution
Given that the area of trapezoid is , , and , we are asked to find the area of triangle where .
Step 1: Area of the Trapezoid
The formula for the area of a trapezoid is: Here, and .
Let the height of the trapezoid be . Then, So, the height of the trapezoid is .
Step 2: Understanding Triangle
Since , triangle and trapezoid share the same height from vertex to line . The height of triangle is thus also .
Let the length of segment .
Step 3: Ratio of Similarity
Since , triangles and are similar. The ratio of similarity is given by: The length of corresponds to the length of .
Step 4: Area of Triangle
The area of triangle is given by:
Step 5: Relationship Between Areas
The area of the trapezoid is composed of the area of and the area of . Because and are similar, the areas of these triangles are proportional to the squares of the corresponding sides: Let , then:
Since represents the proportion of the total area that belongs to and also equals the ratio , we can calculate: Thus:
The correct area calculation is thus: cm².
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Geometry
Similarity
Area Calculation
Formulas
Area of Trapezoid
Area of Triangle
Theorems
Properties of Similar Triangles
Suitable Grade Level
Grades 7-9
Related Recommendation
Calculating Trapezoid Area and Side Length Using Similar Triangles
Find the Face of the Equilateral Trapezoid ABCD: Area Calculation
Calculating Areas of Triangles and Trapezoid in Trapezoid ABCD
Geometry Problem: Area and Perimeter of a Trapezoid with Equal Non-Parallel Sides
Calculate the Area of a Trapezoid with Given Dimensions