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42. using midpoints in the diagram below, b is the mid - point of $over…

Question

  1. using midpoints in the diagram below, b is the mid - point of $overline{ac}$, $ab = 9$, and $ad = 25$. find $cd$.
  2. challenge the mid - point of $overline{ab}$ is $m(7,6)$. the coordinates of point $a$ are $(4,1)$. find the coordinates of point $b$. explain.
  3. multiple choice $t$ is the mid - point of $overline{qr}$. what is the value of $x$? (a) 17 (b) 22 (c) 29.5 (d) 80
  4. multiple choice what is the mid - point of the segment joining $(2,7)$ and $(-6,2)$? (f) $(-2,\frac{9}{2})$ (g) $(-4,0)$ (h) $(-2,4)$ (j) $(\frac{9}{2},-2)$

mixed review
evaluating statements use the diagram at the right to determine whether the statement is true or false. (lessons 1.3, 1.6)

  1. point $a$ lies on line $m$.
  2. point $e$ lies on line $l$.
  3. points $b$, $e$, and $c$ are collinear.
  4. lines $l$ and $m$ intersect at point $b$.
  5. point $e$ is between points $b$ and $c$.
  6. point $f$ is between points $a$ and $b$.

classifying angles name the vertex and sides of the angle. then state whether it appears to be acute, right, obtuse, or straight. (lesson 1.6)
52.
53.
54.
algebra skills
evaluating expressions evaluate the expression. (skills review, p. 670)

  1. $2cdot15 + 40$
  2. $120-35cdot3$
  3. $\frac{1}{2}cdot50 + 145$
  4. $\frac{5}{4}cdot16-20$
  5. $6 + 3cdot5-2$
  6. $11cdot4 + 7-20$
  7. $12cdot2-3cdot4$
  8. $5-10cdot6 + 1$
  9. $2-(3 + 4)cdot5$

Explanation:

Step1: Solve problem 42

Since \(B\) is the mid - point of \(\overline{AC}\) and \(AB = 9\), then \(AC=2AB = 18\). Given \(AD = 25\), by the segment addition postulate \(CD=AD - AC\). So \(CD=25 - 18=7\).

Step2: Solve problem 43

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Let \(A(4,1)\) and \(M(7,6)\), and \(B(x,y)\). Then \(\frac{4 + x}{2}=7\) and \(\frac{1 + y}{2}=6\). Solving \(\frac{4 + x}{2}=7\) gives \(4+x = 14\), so \(x = 10\). Solving \(\frac{1 + y}{2}=6\) gives \(1 + y=12\), so \(y = 11\). The coordinates of \(B\) are \((10,11)\).

Step3: Solve problem 44

Since \(T\) is the mid - point of \(\overline{QR}\), then \(QT=TR\). So \(4x-10 = 78\). Add 10 to both sides: \(4x=78 + 10=88\). Divide both sides by 4: \(x = 22\).

Step4: Solve problem 45

The mid - point formula for two points \((x_1,y_1)=(2,7)\) and \((x_2,y_2)=(-6,2)\) is \(M(\frac{2+( - 6)}{2},\frac{7 + 2}{2})=M(\frac{-4}{2},\frac{9}{2})=M(-2,\frac{9}{2})\).

Step5: Solve problem 46 - 51

Based on the given diagram:

  • For 46: Point \(A\) does not lie on line \(m\), so it's false.
  • For 47: Point \(B\) lies on line \(l\), so it's true.
  • For 48: Points \(B\), \(E\), and \(C\) are not collinear, so it's false.
  • For 49: Lines \(l\) and \(m\) intersect at point \(D\), not \(B\), so it's false.
  • For 50: Point \(E\) is not between points \(B\) and \(C\), so it's false.
  • For 51: Point \(F\) is between points \(A\) and \(B\), so it's true.

Step6: Solve problem 52 - 54

  • For 52: The vertex is \(B\), the sides are \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\), and the angle \(\angle ABC\) appears to be acute.
  • For 53: The vertex is \(J\), the sides are \(\overrightarrow{JK}\) and \(\overrightarrow{JL}\), and the angle \(\angle KJL\) appears to be obtuse.
  • For 54: The vertex is \(Q\), the sides are \(\overrightarrow{QP}\) and \(\overrightarrow{QR}\), and the angle \(\angle PQR\) appears to be straight.

Step7: Solve problem 55 - 63

  • For 55: \(2\times15 + 40=30 + 40=70\).
  • For 56: \(120-35\times3=120 - 105 = 15\).
  • For 57: \(\frac{1}{2}\times50+145=25 + 145=170\).
  • For 58: \(\frac{5}{4}\times16-20=20 - 20=0\).
  • For 59: \(6 + 3\times5-2=6 + 15-2=19\).
  • For 60: \(11\times4+7 - 20=44+7 - 20=31\).
  • For 61: \(12\times2-3\times4=24 - 12=12\).
  • For 62: \(5-10\times6 + 1=5-60 + 1=-54\).
  • For 63: \(2-(3 + 4)\times5=2-7\times5=2 - 35=-33\).

Answer:

  • 42: \(7\)
  • 43: \((10,11)\)
  • 44: \(22\)
  • 45: \((-2,\frac{9}{2})\)
  • 46: False
  • 47: True
  • 48: False
  • 49: False
  • 50: False
  • 51: True
  • 52: Vertex: \(B\), Sides: \(\overrightarrow{BA},\overrightarrow{BC}\), Acute
  • 53: Vertex: \(J\), Sides: \(\overrightarrow{JK},\overrightarrow{JL}\), Obtuse
  • 54: Vertex: \(Q\), Sides: \(\overrightarrow{QP},\overrightarrow{QR}\), Straight
  • 55: \(70\)
  • 56: \(15\)
  • 57: \(170\)
  • 58: \(0\)
  • 59: \(19\)
  • 60: \(31\)
  • 61: \(12\)
  • 62: \(-54\)
  • 63: \(-33\)