Question

25,005 Learners found this answer helpful

name period date
part ii - constructed response
find the value of x in each right triangle. if necessary, round to the nearest tenth.

1. 2.
2. jovan is building a fort in his backyard using a 10 - foot board. he leans the base of the board 6 feet away from the base of the wall. how high will the board be able to reach up the wall? if necessary, round to the nearest tenth.
3. alpia biked 1 mile north from her home to the museum. she then biked 2 miles east from the museum to her friends house. how far will she bike in miles if she bikes home from her friends house in a straight line? if necessary, round to the nearest tenth.
4. a box measures 15 inches by 8 inches by 5 inches. what is the length of its longest diagonal? round to the nearest whole number.

unit 2 - the pythagorean theorem

Explanation:

Step1: Apply Pythagorean theorem ($a^{2}+b^{2}=c^{2}$)

For the first right - triangle, where \(a = 3\), \(c=5\), and \(b=x\).

$$3^{2}+x^{2}=5^{2}$$

$$9 + x^{2}=25$$

Step2: Solve for \(x\)

$$x^{2}=25 - 9$$

$$x^{2}=16$$

$$x=\sqrt{16}$$

$$x = 4$$

Step3: Apply Pythagorean theorem for the second right - triangle

Where \(a = 8\), \(b = 6\), and \(c=x\)

$$8^{2}+6^{2}=x^{2}$$

$$64+36=x^{2}$$

$$x^{2}=100$$

$$x=\sqrt{100}$$

$$x = 10$$

Step4: For the third problem (Jovan's board)

Let the height be \(h\), the length of the board \(c = 10\) (hypotenuse), and the base \(a=6\)

$$h^{2}+6^{2}=10^{2}$$

$$h^{2}+36 = 100$$

$$h^{2}=100 - 36$$

$$h^{2}=64$$

$$h=\sqrt{64}$$

$$h = 8$$

Step5: For the fourth problem (Alpia's bike ride)

The two legs of the right - triangle are \(a = 1\) (north) and \(b = 2\) (east). Let the distance she bikes home be \(d\)

$$d^{2}=1^{2}+2^{2}$$

$$d^{2}=1 + 4$$

$$d^{2}=5$$

$$d=\sqrt{5}\approx2.2$$

Step6: For the fifth problem (box diagonal)

First, find the diagonal of the base (\(d_{1}\)): if the base has sides \(l = 15\) and \(w = 8\)

$$d_{1}^{2}=15^{2}+8^{2}$$

$$d_{1}^{2}=225+64$$

$$d_{1}^{2}=289$$

$$d_{1}=17$$

Then find the space diagonal (\(D\)) with height \(h = 5\)

$$D^{2}=d_{1}^{2}+h^{2}$$

$$D^{2}=17^{2}+5^{2}$$

$$D^{2}=289 + 25$$

$$D^{2}=314$$

$$D=\sqrt{314}\approx18$$

Answer:

1. \(x = 4\)
2. \(x = 10\)
3. The board will reach \(8\) feet high.
4. She will bike approximately \(2.2\) miles.
5. The length of the longest diagonal is approximately \(18\) inches.