Question

The inradius of a right angled triangle with sides \(\mathrm{a}, \mathrm{b}, \mathrm{c}\), where \(\mathrm{c}\) is the hypotenuse is 5 . Then, (a) \(a+b-c=10\) (b) \(\mathrm{ab}+10 \mathrm{c}=50\) (c) \(\quad a b-10 c=50\) (d) \(r=\frac{5 \sin A}{\sqrt{2}}\)

Short answer

(a) a+b-c=10 (b) ab+10c=50 (c) ab-10c=50 (d) r = 5 * sinA /√2 Answer: (a) and (d) are true.

Step-by-step solution

Relationship between inradius, semiperimeter, and area of a triangle

The relationship between the inradius (r), semiperimeter (s), and area (A) of a triangle can be expressed as follows: \(r = \frac{A}{s}\)

Inradius and semiperimeter formulas for a right-angled triangle

To find the inradius and semiperimeter of a right-angled triangle, we need the following formulas: 1. Inradius r: \(r = \frac{a + b - c}{2}\) 2. Semiperimeter s: \(s = \frac{a + b + c}{2}\).

Area formula for a right-angled triangle

The area of a right-angled triangle can be calculated using the formula: \(A = \frac{1}{2}ab\).

Checking validity of each option

Now, we will use the formulas and relationships from Steps 1-3 to evaluate each option: (a) \(a+b-c=10\) Using the inradius formula, \(r = \frac{a + b - c}{2}\) Since the inradius r is given as 5, \(5 = \frac{a + b - c}{2}\) Multiplying by 2, \(a+b-c=10\) So, option (a) is true. (b) \(ab+10c=50\) Using the area formula, \(A = \frac{1}{2}ab\) Since we know from Step 1 that r = 5 and using the semiperimeter formula: \(r = \frac{A}{s} \Rightarrow 5 = \frac{\frac{1}{2}ab}{\frac{a+b+c}{2}} \Rightarrow 5\left(\frac{a+b+c}{2}\right)=\frac{1}{2}ab\) \(5(a + b + c)=ab\) So, option (b) is not true, as it is not the same as the equation we derived. (c) \(ab-10c=50\) As we derived the correct equation for option (b) as \(5(a + b + c)=ab\), option (c) is also not true. (d) \(r=\frac{5 \sin A}{\sqrt{2}}\) Here, we derive the formula for the inradius using trigonometry. Since it’s a right-angled triangle, we have: \(r = \frac{a}{2} \sin A\) Given \(r = 5\), we get \(5 = \frac{a}{2} \sin A \Rightarrow r = \frac{5 \sin A}{\sqrt{2}}\) So, option (d) is also true. Therefore, the correct solutions are options (a) and (d).

Key concepts

Semiperimeter of a Triangle

Understanding the semiperimeter of a triangle is essential when solving problems that involve the inradius, such as the area or other related properties.

In its simplest form, the semiperimeter, often represented by the symbol 's', is just half of the perimeter of the triangle. It is calculated by adding up the lengths of all three sides of the triangle and dividing by two, or in formula form: \[s = \frac{a + b + c}{2}\].

The semiperimeter is a very useful concept as it plays a key role in various mathematical formulas, including Heron's formula for the area of a triangle, and the relationship between the inradius and the area of a triangle.

Area of a Right-Angled Triangle

Calculating the area of a right-angled triangle is one of the fundamental concepts in geometry. Unlike other triangles, for a right-angled triangle, it's relatively easier because one angle is always 90 degrees, which defines the base and height clearly.

The area of a right-angled triangle is found using the formula: \[A = \frac{1}{2}ab\], where 'a' and 'b' are the two sides that form the right angle, often referred to as the 'base' and 'height' respectively. This formula is a specific case of the general formula for the area of a triangle (\[A = \frac{1}{2} \times base \times height\]) where the base and height are perpendicular to each other.

This concept is not only crucial in pure mathematics but also finds its application in various real-world problems that involve right-angled shapes, such as in construction and carpentry.

Trigonometry in Triangles

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. In the context of right-angled triangles, trigonometry provides a set of functions - sine (sin), cosine (cos), and tangent (tan) - which are essential to solving many types of geometry problems.

For example, to find the length of a side or to determine the size of an angle, you can use these trigonometric functions. The ratios are defined as: \[\sin(\theta) = \frac{opposite}{hypotenuse}\], \[\cos(\theta) = \frac{adjacent}{hypotenuse}\], \[\tan(\theta) = \frac{opposite}{adjacent}\].

These functions are particularly useful when dealing with the properties of a right-angled triangle, which is governed by the Pythagorean theorem. By using trigonometry, we can interrelate the inradius, the sides of the triangle, and the angles, providing a more comprehensive understanding of the triangle's geometry. For instance, in the provided exercise, the inradius is connected with the angle A through the sine function to validate one of the options.