$$**Question:** A circle is inscribed in a right triangle with legs of lengths 9 cm and 12 cm. Determine the radius of the inscribed circle.

$$**Question:** A circle is inscribed in a right triangle with legs of lengths 9 cm and 12 cm. Determine the radius of the inscribed circle.

["Title: How to Find the Radius of the Inscribed Circle in a Right Triangle – Step-by-Step Guide", "---", "Question: A circle is inscribed in a right triangle with legs of lengths 9 cm and 12 cm. Determine the radius of the inscribed circle.", "If you’ve ever wondered how to find the radius of a circle perfectly fitted inside a right triangle—especially one with legs measuring 9 cm and 12 cm—you’ve come to the right place. This elegant geometric configuration reveals a simple yet powerful formula to calculate the radius of an inscribed circle, making it easier to solve similar triangle problems efficiently.", "---", "### Understanding the Problem", "We’re given a right triangle with legs of 9 cm and 12 cm. The hypotenuse can be quickly calculated using the Pythagorean theorem:", "[\nc = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \ ext{ cm}\n]", "Now, the challenge is to determine the radius ( r ) of the circle inscribed within this triangle—meaning the circle tangent to all three sides from the inside.", "---", "### Formula for the Radius of an Inscribed Circle in a Right Triangle", "For any right triangle, the radius ( r ) of the incircle is given by:", "[\nr = \frac{a + b - c}{2}\n]", "Where:\n- ( a ) and ( b ) are the legs\n- ( c ) is the hypotenuse", "Alternatively, a more widely used formula from triangle geometry is:", "[\nr = \frac{\ ext{Area}}{\ ext{Semiperimeter}}\n]", "Let’s explore both to clearly see how the radius is found.", "---", "### Method 1: Using the Formula ( r = \frac{a + b - c}{2} )", "Plug in the values:\n( a = 9 ), ( b = 12 ), ( c = 15 ):", "[\nr = \frac{9 + 12 - 15}{2} = \frac{6}{2} = 3 \ ext{ cm}\n]", "---", "### Method 2: Using Area and Semiperimeter", "First, compute the area ( A ) of the triangle:", "[\nA = \frac{1}{2} \ imes 9 \ imes 12 = 54 \ ext{ cm}^2\n]", "Next, compute the semiperimeter ( s ):", "[\ns = \frac{a + b + c}{2} = \frac{9 + 12 + 15}{2} = \frac{36}{2} = 18 \ ext{ cm}\n]", "Then, use ( r = \frac{A}{s} ):", "[\nr = \frac{54}{18} = 3 \ ext{ cm}\n]", "---", "### Conclusion", "Both methods confirm that the radius of the inscribed circle in a right triangle with legs 9 cm and 12 cm is 3 cm. This elegant solution reflects the harmony of geometry, offering a quick and reliable approach for right triangles.", "Next time you encounter a problem involving an inscribed circle in a right triangle, apply the formula confidently:", "[\n\boxed{r = \frac{a + b - c}{2} \quad \ ext{or} \quad r = \frac{A}{s}}\n]", "---", "Keywords: inscribed circle radius, right triangle incircle, formula for incircle of right triangle, calculate incircle radius, 9 cm 12 cm triangle, triangle geometry tip, how to find radius of incircle", "Meta description: Learn the exact method to calculate the radius of an inscribed circle in a right triangle with legs 9 cm and 12 cm using geometric formulas and step-by-step calculations.", "---", "Ready to dive deeper into geometric formulas? Explore more tips on inscribed circles, tangents, and triangle proportions in our full geometry guide!"]

Related Articles

Trending Articles

During a 7-day photography expedition in Tanzania, your team captures 120 images of elephants, 85 images of lions, and 45 images of rhinoceroses. Due to a technical error, you lose 15% of the total images. How many total images remain after the loss?First, calculate the total images: 120 + 85 + 45 = <<120+85+45=250>>250 imagesBioinformatician Analyzing Genetic Sequences:A genomic dataset contains 1.2 million DNA sequences. An algorithm filters out 38% of low-quality sequences, then removes another 12% of the remaining sequences as redundant. How many sequences are left after both filters?You plan a 10-day trek to photograph giraffes, spending 60% of the days in the Serengeti, 20% in Maasai Mara, and the remaining days in Ngorongoro. Due to weather, you only complete 75% of the planned Serengeti days. How many actual Serengeti days did you photograph?A genome has 3.2 billion base pairs. A sequencing error introduces an average mutation in 0.007% of bases. How many errors are expected in the full genome?Convert percentage: 0.007% = 0.00007A dataset has 450,000 gene expressions. A normalization algorithm reduces variance in 28% of genes, and a subsequent quality flag removes 15% of the remaining genes as non-expressing. How many genes pass all filters?A machine learning model predicts gene functions with 94% accuracy on 2.5 million annotations. How many predictions are incorrect?Over 12 days, you photograph 52 lion prides. Each pride has a median of 14 members, but 12.5% of prides are temporarily dispersed and not photographed. On average, each photographed pride contributes 11 usable images. How many usable images are collected?