Try this beautiful problem from Geometry based on Radius of a semicircle inscribed in an isosceles triangle.
A semicircle is inscribed in an isoscles triangle with base 16 and height 15 so that the diameter of the semicircle is contained in the base of the triangle as shown .what is the radius of the semicircle?
Geometry
Area
pythagoras
Answer:$\frac{120}{17}$
AMC-8, 2016 problem 25
Challenges and Thrills of Pre College Mathematics
Draw a perpendicular from the point C on base AB
Can you now finish the problem ..........
D be the midpoint of the AB(since $\triangle ABC $ is an isoscles Triangle)
Find AC and area
can you finish the problem........
Area of the $\triangle ABC= \frac{1}{2} \times AB \times CD$
= $ \frac{1}{2} \times 16 \times 15 $
=120 sq.unit
Using the pythagoras th. $ AC^2= AD^2+CD^2$
i.e $AC^2=(8)^2+(15)^2$
i.e $AC=17$
Let$ ED = x$ be the radius of the semicircle
Therefore Area of $\triangle CAD = \frac{1}{2} \times AC \times ED$=$\frac {1}{2} area of \triangle ABC$
i.e $\frac{1}{2} \times AC \times ED $=60
i.e $\frac{1}{2} \times 17 \times x$ =60
i.e $x=\frac {120}{7}$
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
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 […]