Question

name:
section:

1. the coordinates of the endpoints (overline{qr}) are q(8,2) and r(5,7). find the

is closest to the length of (overline{qr}) in units. round to the nearest tenths if necessary.

1. find the missing angle. write an equation, then solve for x. remember to include the

units.
a)
b)
c)
d)

Explanation:

Problem 15: Length of Segment \( \overline{QR} \)

To find the length of a segment with endpoints \( Q(x_1, y_1) \) and \( R(x_2, y_2) \), we use the distance formula:

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Step 1: Identify Coordinates

For \( Q(8, 2) \) and \( R(5, 7) \), we have:
\( x_1 = 8, \, y_1 = 2 \)
\( x_2 = 5, \, y_2 = 7 \)

Step 2: Substitute into Distance Formula

Calculate the differences in coordinates:
\( x_2 - x_1 = 5 - 8 = -3 \)
\( y_2 - y_1 = 7 - 2 = 5 \)

Square these differences:
\( (-3)^2 = 9 \)
\( 5^2 = 25 \)

Sum the squares:
\( 9 + 25 = 34 \)

Take the square root:
\( d = \sqrt{34} \approx 5.8 \) (rounded to the nearest tenth)

Problem 16: Missing Angles in Triangles

Part (a): Triangle with Angles \( 37^\circ \), \( 66^\circ \), and \( x^\circ \)

The sum of angles in a triangle is \( 180^\circ \).

Step 1: Write the Equation

\( 37 + 66 + x = 180 \)

Step 2: Solve for \( x \)

Simplify the left side:
\( 103 + x = 180 \)

Subtract 103 from both sides:
\( x = 180 - 103 = 77 \)

Part (b): Right Triangle with Angles \( 48^\circ \), \( 90^\circ \), and \( x^\circ \)

The sum of angles in a triangle is \( 180^\circ \).

Step 1: Write the Equation

\( 48 + 90 + x = 180 \)

Step 2: Solve for \( x \)

Simplify the left side:
\( 138 + x = 180 \)

Subtract 138 from both sides:
\( x = 180 - 138 = 42 \)

Part (c): Triangle with Angles \( 5x^\circ \), \( 12x^\circ \), and \( 85^\circ \)

The sum of angles in a triangle is \( 180^\circ \).

Step 1: Write the Equation

\( 5x + 12x + 85 = 180 \)

Step 2: Combine Like Terms

\( 17x + 85 = 180 \)

Step 3: Solve for \( x \)

Subtract 85 from both sides:
\( 17x = 180 - 85 = 95 \)

Divide by 17:
\( x = \frac{95}{17} \approx 5.6 \) (rounded to the nearest tenth)

Part (d): Right Triangle with Angles \( 42^\circ \), \( 90^\circ \), and \( 6x^\circ \)

The sum of angles in a triangle is \( 180^\circ \).

Step 1: Write the Equation

\( 42 + 90 + 6x = 180 \)

Step 2: Simplify

\( 132 + 6x = 180 \)

Step 3: Solve for \( x \)

Subtract 132 from both sides:
\( 6x = 180 - 132 = 48 \)

Divide by 6:
\( x = \frac{48}{6} = 8 \)

Answer:

s:

1. Length of \( \overline{QR} \approx \boldsymbol{5.8} \) units.

16.
a) \( x = \boldsymbol{77^\circ} \)
b) \( x = \boldsymbol{42^\circ} \)
c) \( x \approx \boldsymbol{5.6^\circ} \) (or \( x = \frac{95}{17} \))
d) \( x = \boldsymbol{8^\circ} \)