Example 2: Triangle DEF .Given: \( \circ \) Hypotenuse \( \mathrm{DE}=13 \mathrm{~cm} \). \( \circ \) One leg \( \mathrm{DF}=5 \mathrm{~cm} \). \( \circ \angle \mathrm{F}=90 \)

To solve the problem involving triangle DEF, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is given by: \[ c^2 = a^2 + b^2 \] In this case: - The hypotenuse \( DE = 13 \, \text{cm} \) (which we will denote as \( c \)). - One leg \( DF = 5 \, \text{cm} \) (which we will denote as \( a \)). - We need to find the length of the other leg \( EF \) (which we will denote as \( b \)). ### Step 1: Set up the equation Using the Pythagorean theorem, we can set up the equation as follows: \[ DE^2 = DF^2 + EF^2 \] Substituting the known values: \[ 13^2 = 5^2 + EF^2 \] ### Step 2: Calculate the squares Calculating the squares: \[ 169 = 25 + EF^2 \] ### Step 3: Solve for \( EF^2 \) Now, we can isolate \( EF^2 \): \[ EF^2 = 169 - 25 \] Calculating the right side: \[ EF^2 = 144 \] ### Step 4: Find \( EF \) Now, we take the square root of both sides to find \( EF \): \[ EF = \sqrt{144} \] Calculating the square root: \[ EF = 12 \, \text{cm} \] ### Conclusion The length of the other leg \( EF \) is \( 12 \, \text{cm} \). Thus, the dimensions of triangle DEF are: - Hypotenuse \( DE = 13 \, \text{cm} \) - Leg \( DF = 5 \, \text{cm} \) - Leg \( EF = 12 \, \text{cm} \)