Question

1. two side lengths of right triangle abc are given. what is the length of \\( \overline { b c } \\)?

a. 20 cm
b. 36 cm
c. 50 cm
d. 400 cm

1. the distance across a lake cannot be directly measured. a land surveyor uses this map to determine d, the distance across the lake. what is the distance across the lake?

a. 8 kilometers
b. 10 kilometers
c. 12 kilometers
d. 14 kilometers

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. In problem 5, \(AB = 29\) (hypotenuse), \(AC=21\), and we want to find \(BC\). Let \(BC = x\), then \(x^{2}+21^{2}=29^{2}\).

Step2: Calculate \(x^{2}\)

\(x^{2}=29^{2}-21^{2}\). Using the formula \(a^{2}-b^{2}=(a + b)(a - b)\), we have \(x^{2}=(29 + 21)(29-21)=(50)\times(8)=400\).

Step3: Solve for \(x\)

Take the square root of \(x^{2}\), \(x=\sqrt{400}=20\).

For problem 6:

Step1: Apply Pythagorean theorem

Let the distance across the lake be \(d\). The hypotenuse \(c = 13\) km and one leg \(b = 5\) km. Using \(a^{2}+b^{2}=c^{2}\), we get \(d^{2}+5^{2}=13^{2}\).

Step2: Calculate \(d^{2}\)

\(d^{2}=13^{2}-5^{2}\). Using \(a^{2}-b^{2}=(a + b)(a - b)\), \(d^{2}=(13 + 5)(13-5)=(18)\times(8)=144\).

Step3: Solve for \(d\)

Take the square root of \(d^{2}\), \(d=\sqrt{144}=12\).

Answer:

1. A. \(20\) cm
2. C. \(12\) kilometers