Question: Day Four Points B and C lie on segment AD as shown below. Segment AD is 32 units long, segment AC is 23 units long, and segment BD is 27 units long. How many units long is segment BC? A. 21 B. 18 C. 9 D. 4 E. Cannot be determined from the given information. 14. Which point is the third quadrant (x, y) coordinate plane below has the coordinates (2, -5)? A. A B. B C. C D. D E. E Day Five What is the intersection of a non-parallel plane A and line B (that is contained in a separate plane C)? A. line B. line segment C. point D. plane E. angle In the standard (x, y) coordinate plane, the midpoint of AB is (3, 2), and A is located at (0, -7). What are the coordinates of B? F. (-1.5, 0.5) G. (-5, 5) H. (6, -4) J. (2, 3) K. (2, -4) Day Six In the figure shown below, the measure of \u2220SRQ is $ (x + 15)^\circ $ and the measure of \u2220SRU is $ 90^\circ $. What is the measure of \u2220TRU? In the standard (x, y) coordinate plane, point X has coordinates $ (-4, 0) $ and point Y has coordinates $ (0, -8) $. What are the coordinates of the midpoint of XY? A. (-6, -1) B. (-2, -4) C. (0, 2) D. (2, 0) E. (6, -1)

1. **Problem 1: Length of segment BC** Given: Points B and C lie on segment AD. Segment AD = $32$ units, AC = $23$ units, BD = $27$ units. Find: Length of BC. 2. **Understanding the problem:** Since B and C lie on AD, and AD is a straight segment, the points are collinear. We know: $$ AD = AB + BC + CD $$ But we only have AC and BD, so we need to find BC using these. 3. **Express segments in terms of points:** - $AC = AB + BC = 23$ - $BD = BC + CD = 27$ - $AD = AB + BC + CD = 32$ 4. **Set variables:** Let $AB = a$, $BC = b$, $CD = c$. From above: $$ a + b = 23 $$ $$ b + c = 27 $$ $$ a + b + c = 32 $$ 5. **Use the equations:** From the third equation: $$ a + b + c = 32 $$ Substitute $a = 23 - b$ and $c = 27 - b$: $$ (23 - b) + b + (27 - b) = 32 $$ Simplify: $$ 23 - b + b + 27 - b = 32 $$ $$ 50 - b = 32 $$ 6. **Solve for $b$:** $$ 50 - b = 32 $$ $$ b = 50 - 32 = 18 $$ 7. **Answer:** Length of segment BC is $18$ units. --- 8. **Problem 2: Point in the third quadrant with coordinates (2, -5)** Recall: Third quadrant points have both $x < 0$ and $y < 0$. Given point is $(2, -5)$ where $x=2 > 0$, so it is not in the third quadrant. 9. **Answer:** None of the points with coordinates $(2, -5)$ lie in the third quadrant. So the answer is **E. E** (assuming E means none or no correct point). --- 10. **Problem 3: Intersection of a non-parallel plane A and line B (contained in plane C)** - A plane and a line intersect in either a point, line, or no intersection. - Since line B lies in plane C, and plane A is non-parallel to plane C, their intersection is a line. - The intersection of plane A and line B is a point (since line B is contained in plane C, and plane A intersects plane C in a line, the intersection with line B is a point). 11. **Answer:** The intersection is a **point**. --- 12. **Problem 4: Find coordinates of B given midpoint and A** Given midpoint $M = (3, 2)$ and point $A = (0, -7)$. Midpoint formula: $$ M = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) $$ 13. **Set up equations:** $$ 3 = \frac{0 + x_B}{2} \Rightarrow x_B = 6 $$ $$ 2 = \frac{-7 + y_B}{2} \Rightarrow -7 + y_B = 4 \Rightarrow y_B = 11 $$ 14. **Answer:** Coordinates of B are $(6, 11)$, but this is not among the options. Check options again; none match $(6, 11)$. Since none match, possibly a typo or error in options. --- 15. **Problem 5: Measure of angle TRU** Given: $$ \angle SRQ = (x + 15)^\circ $$ $$ \angle SRU = 90^\circ $$ Assuming $\angle TRU$ is complementary or related to these angles, but no further info is given. Cannot determine $\angle TRU$ without more data. --- 16. **Problem 6: Midpoint of XY** Given points: $$ X = (-4, 0), Y = (0, -8) $$ Midpoint formula: $$ M = \left( \frac{-4 + 0}{2}, \frac{0 + (-8)}{2} \right) = (-2, -4) $$ 17. **Answer:** Midpoint is $(-2, -4)$ which corresponds to option B. --- **Final answers:** - Problem 1: BC = $18$ units (B) - Problem 2: No point with (2, -5) in third quadrant (E) - Problem 3: Intersection is a point (C) - Problem 4: Coordinates of B = $(6, 11)$ (not listed) - Problem 5: Cannot determine $\angle TRU$ - Problem 6: Midpoint of XY = $(-2, -4)$ (B)