Question

in a right triangle, if one leg is 24 units and the hypotenuse is 25 units, what is the length of the other leg?
a. 6 units
b. 4 units
c. 7 units
d. 5 units

a soccer field is 100 yards long and 50 yards wide. what is the diagonal distance across the field?
a. 125 yards
b. 111.8 yards
c. 120 yards
d. 110 yards

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\).

Step2: Solve the first problem

Let one leg \(a = 24\), hypotenuse \(c = 25\), and the other leg be \(b\). Then \(b^{2}=c^{2}-a^{2}\). Substitute the values: \(b^{2}=25^{2}-24^{2}=(25 + 24)(25 - 24)=49\times1 = 49\). So \(b=\sqrt{49}=7\) units.

Step3: Solve the second problem

The length of the soccer - field \(a = 100\) yards and width \(b = 50\) yards. The diagonal \(d\) is the hypotenuse of a right - triangle. Using the Pythagorean theorem \(d^{2}=a^{2}+b^{2}\), so \(d^{2}=100^{2}+50^{2}=10000 + 2500=12500\). Then \(d=\sqrt{12500}=50\sqrt{5}\approx50\times2.236 = 111.8\) yards.

Answer:

c. 7 units
b. 111.8 yards