middle of topic quiz\npythagorean theorem, number sets\n1. true or false?\nplace an x in the appropriate…

middle of topic quiz\npythagorean theorem, number sets\n1. true or false?\nplace an x in the appropriate space. (1 mark each)\ntrue / false\na) integers are a subset/part of irrational numbers.\nb) $\\frac{1}{33}=0.0303$\nc) $\\frac{7}{19}$ is rational.\nd) $\\sqrt{256}$ is rational.\ne) $\\pi = \\frac{22}{7}$\nmultiple choice: (2 marks each)\n2. a farmer planted wheat in a triangular field. this field is represented by $\\triangle abc$, where $m\\overline{ab}=8.5$ km, $m\\overline{bc}=5$ km and $m\\overline{dc}=3$ km\nwhat is the area of the entire field?\na) $12.75$ km²\nb) $17$ km²\nc) $21$ km²\nd) $36$ km²\n3. which algebraic expression is true for the given right - triangle? pay attention to brackets.\na) $4y^{2}=5x^{2}-3z^{2}$\nb) $\\sqrt{25x^{2}-9z^{2}} = 4y$\nc) $\\sqrt{16y^{2}+9z^{2}} = 25x^{2}$\nd) $5^{2}x = 3^{2}z+4^{2}y$

middle of topic quiz\npythagorean theorem, number sets\n1. true or false?\nplace an x in the appropriate space. (1 mark each)\ntrue / false\na) integers are a subset/part of irrational numbers.\nb) $\\frac{1}{33}=0.0303$\nc) $\\frac{7}{19}$ is rational.\nd) $\\sqrt{256}$ is rational.\ne) $\\pi = \\frac{22}{7}$\nmultiple choice: (2 marks each)\n2. a farmer planted wheat in a triangular field. this field is represented by $\\triangle abc$, where $m\\overline{ab}=8.5$ km, $m\\overline{bc}=5$ km and $m\\overline{dc}=3$ km\nwhat is the area of the entire field?\na) $12.75$ km²\nb) $17$ km²\nc) $21$ km²\nd) $36$ km²\n3. which algebraic expression is true for the given right - triangle? pay attention to brackets.\na) $4y^{2}=5x^{2}-3z^{2}$\nb) $\\sqrt{25x^{2}-9z^{2}} = 4y$\nc) $\\sqrt{16y^{2}+9z^{2}} = 25x^{2}$\nd) $5^{2}x = 3^{2}z+4^{2}y$

Answer

1.

a.

정수는 무리수 집합의 부분집합이 아니다. 정수는 유리수이다. 따라서 False.

b.

$\frac{1}{33}=0.\overline{03}\neq0.0303$. 따라서 False.

c.

유리수는 $\frac{p}{q}$ ($p,q$는 정수, $q\neq0$) 꼴의 수이다. $\frac{7}{19}$는 $\frac{p}{q}$ 꼴이므로 유리수이다. 따라서 True.

d.

$\sqrt{256} = 16$, 16은 정수이므로 유리수이다. 따라서 True.

e.

$\pi$는 무리수이며, $\frac{22}{7}$은 유리수이다. $\pi\neq\frac{22}{7}$. 따라서 False.

2.

Step1: 높이 찾기

삼각형 $ABC$에서 $AD$를 높이라고 가정하자. $BD = BC - DC=5 - 3 = 2$km. $AD=\sqrt{AB^{2}-BD^{2}}$ (피타고라스 정리). $AB = 8.5$km, $BD = 2$km이므로, $AD=\sqrt{8.5^{2}-2^{2}}=\sqrt{72.25 - 4}=\sqrt{68.25}$km.

Step2: 넓이 계산

삼각형 넓이 공식 $S=\frac{1}{2}\times밑변\times높이$. 밑변 $BC = 5$km, 높이 $AD=\sqrt{68.25}$km이므로, $S=\frac{1}{2}\times5\times\sqrt{68.25}\approx\frac{1}{2}\times5\times 8.26 = 20.65\approx21$km². 따라서 C.

3.

피타고라스 정리에 의하면, 직각삼각형에서 빗변의 제곱은 다른 두 변의 제곱의 합이다. $(5x)^{2}=(4y)^{2}+(3z)^{2}$ $25x^{2}=16y^{2}+9z^{2}$ $\sqrt{25x^{2}-9z^{2}} = 4y$. 따라서 B.

Answer:

  1. a. False, b. False, c. True, d. True, e. False
  2. C. $21$ km²
  3. B. $\sqrt{25x^{2}-9z^{2}} = 4y$