A right triangle has a hypotenuse of length \( z = 10 \) cm and an inscribed circle with radius \( c = 2 \) cm. What is the ratio of the area of the circle to the area of the triangle?
["Right Triangle with Hypotenuse ( z = 10 ) cm and Inscribed Circle Radius ( c = 2 ) cm: Find the Area Ratio", "Understanding the geometry of right triangles with inscribed circles (incircles) is key to solving problems involving area comparisons. In this article, we explore a specific right triangle where the hypotenuse ( z = 10 ) cm and the radius of the inscribed circle is ( c = 2 ) cm. We’ll calculate the triangle’s area, determine the area of the incircle, and find the precise ratio of the circle’s area to the triangle’s area—helpful for math learners, educators, and puzzle enthusiasts.", "---", "### What is an Inscribed Circle in a Right Triangle?", "An inscribed circle touches all three sides of a triangle, fitting perfectly within it. For a right triangle, the inradius ( c ) can be related directly to the triangle’s legs ( a ), ( b ), and hypotenuse ( z ) via a known formula:", "[\nc = \frac{a + b - z}{2}\n]", "Given:\n- ( z = 10 ) cm\n- ( c = 2 ) cm", "Substitute into the formula:", "[\n2 = \frac{a + b - 10}{2}\n]", "Multiply both sides by 2:", "[\n4 = a + b - 10 \quad \Rightarrow \quad a + b = 14\n]", "---", "### Use the Pythagorean Theorem and Area Relations", "We know from the Pythagorean theorem:", "[\na^2 + b^2 = z^2 = 100\n]", "We also know:", "[\na + b = 14\n]", "To find ( a ) and ( b ), square the sum:", "[\n(a + b)^2 = 14^2 = 196\n]", "Expand:", "[\na^2 + 2ab + b^2 = 196\n]", "Substitute ( a^2 + b^2 = 100 ):", "[\n100 + 2ab = 196 \quad \Rightarrow \quad 2ab = 96 \quad \Rightarrow \quad ab = 48\n]", "---", "### Calculate Areas", "Now compute the area of the right triangle:", "[\n\ ext{Area}{\ riangle} = \frac{1}{2}ab = \frac{1}{2} \ imes 48 = 24 \ ext{ cm}^2\n]", "Area of the inscribed circle:", "[\n\ ext{Area}^2}} = \pi c^2 = \pi \ imes 2^2 = 4\pi \ ext{ cm\n]", "---", "### Compute the Ratio", "The ratio of the circle’s area to the triangle’s area is:", "[\n\ ext{Ratio} = \frac{\ ext{Area}{\ ext{circle}}}{\ ext{Area}}} = \frac{4\pi}{24} = \frac{\pi}{6\n]", "---", "### Final Answer", "For a right triangle with hypotenuse ( z = 10 ) cm and inradius ( c = 2 ) cm, the ratio of the area of the inscribed circle to the area of the triangle is:", "[\n\boxed{\frac{\pi}{6}}\n]", "This elegant result connects algebraic geometry with real-world circle and triangle relationships. Whether you're solving problems in trigonometry, designing geometric models, or studying mathematical proofs, understanding how inradius influences area provides powerful insight.", "---", "Keywords: right triangle, hypotenuse, inscribed circle, inradius, area ratio, right triangle geometry, incircle area formula, ( z = 10 ), ( c = 2 ), ( \pi/6 )"]
