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
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.
Cheenta students shine at the Purple Comet Math Meet 2025 organized by Titu Andreescu and Jonathan Kanewith top national and global ranks.
Celebrate the success of Cheenta students in the Stanford Math Tournament. The Unified Vectors team achieved Top 20 in the Team Round.