If \( \triangle A B C \) is right at \( \angle B \& b=10 \mathrm{~cm} \) then \( \frac{a}{a+b}+\frac{c}{\sin c}=\ldots \ldots \ldots \) \( \begin{array}{llll}\text { a) } 10 & \text { b) } 20 & \text { c) } 100 & \text { d) } 40\end{array} \)

In the right triangle \( \triangle ABC \) with \( \angle B = 90^\circ \), we have the following sides: - \( a \) is the length of side opposite angle \( A \). - \( b = 10 \, \text{cm} \) is the length of side opposite angle \( C \). - \( c \) is the hypotenuse. Using the Pythagorean theorem, we have: \[ c^2 = a^2 + b^2 \] Now, we need to evaluate the expression: \[ \frac{a}{a+b} + \frac{c}{\sin C} \] First, we need to find \( \sin C \). In a right triangle, \( \sin C \) is given by: \[ \sin C = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c} = \frac{10}{c} \] Now substituting this into the expression: \[ \frac{c}{\sin C} = \frac{c}{\frac{10}{c}} = \frac{c^2}{10} \] Now we can rewrite the entire expression: \[ \frac{a}{a+b} + \frac{c^2}{10} \] Next, we need to express \( c^2 \) in terms of \( a \) and \( b \): \[ c^2 = a^2 + b^2 = a^2 + 10^2 = a^2 + 100 \] Substituting this back into our expression gives: \[ \frac{a}{a+b} + \frac{a^2 + 100}{10} \] Now, we can simplify this expression. However, we need to find a specific value for \( \frac{a}{a+b} + \frac{c^2}{10} \). To find \( a \), we can use the fact that \( a \) and \( b \) are related through the angles in the triangle. However, without loss of generality, we can assume \( a \) is some value that satisfies the triangle's properties. Let's assume \( a = 10 \, \text{cm} \) (for simplicity): 1. Then \( b = 10 \, \text{cm} \). 2. Calculate \( c \): \[ c^2 = a^2 + b^2 = 10^2 + 10^2 = 100 + 100 = 200 \implies c = \sqrt{200} = 10\sqrt{2} \] Now substituting \( a = 10 \) into the expression: \[ \frac{10}{10 + 10} + \frac{(10\sqrt{2})^2}{10} = \frac{10}{20} + \frac{200}{10} = \frac{1}{2} + 20 = 20.5 \] However, we need to check the options given: - a) 10 - b) 20 - c) 100 - d) 40 Since \( 20.5 \) does not match any of the options, we need to check our assumptions or calculations. Let's try \( a = 0 \) (the minimum case): 1. Then \( b = 10 \, \text{cm} \). 2. Calculate \( c \): \[ c^2 = 0^2 + 10^2 = 0 + 100 = 100 \implies c = 10 \] Substituting \( a = 0 \): \[ \frac{0}{0 + 10} + \frac{10}{\sin C} = 0 + \frac{10}{1} = 10 \] Thus, the answer is: \[ \boxed{10} \]