Try this beautiful problem from Geometry based on Area of Trapezoid.
The area of triangle XYZ is 8 square inches. Points A and B are midpoints of congruent segments XY and XZ . Altitude XC bisects YZ.What is the area (in square inches) of the shaded region?
Geometry
Triangle
Trapezoid
Answer:\(3\)
AMC-8 (2002) Problem 20
Pre College Mathematics
Given that Points A and B are midpoints of congruent segments XY and XZ and Altitude XC bisects YZ
Let us assume that the length of YZ=\(x\) and length of \(XC\)= \(y\)
Can you now finish the problem ..........
Therefore area of the trapezoid= \(\frac{1}{2} \times (YC+AO) \times OC\)
can you finish the problem........
Let us assume that the length of YZ=\(x\) and length of \(XC\)= \(y\)
Given that area of \(\triangle xyz\)=8
Therefore \(\frac{1}{2} \times YZ \times XC\)=8
\(\Rightarrow \frac{1}{2} \times x \times y\) =8
\(\Rightarrow xy=16\)
Given that Points A and B are midpoints of congruent segments XY and XZ and Altitude XC bisects YZ
Then by the mid point theorm we can say that \(AO=\frac{1}{2} YC =\frac{1}{4} YZ =\frac{x}{4}\) and \(OC=\frac{1}{2} XC=\frac{y}{2}\)
Therefore area of the trapezoid shaded area = \(\frac{1}{2} \times (YC+AO) \times OC\)= \(\frac{1}{2} \times ( \frac{x}{2} + \frac{x}{4} ) \times \frac{y}{2}\) =\(\frac{3xy}{16}=3\) (as \(xy\)=16)
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.