اختبار اختيار من متعدد تفاعلي: Geometry Practice: Segment Addition and Congruence
هذا الاختبار مخصص لممارسة مهارات الهندسة المتعلقة بمسلمة جمع القطع المستقيمة وخصائص التطابق. يتضمن الاختبار مجموعة من الأسئلة التي تتطلب حل معادلات جبرية لإيجاد قيم المتغيرات وأطوال القطع المستقيمة الموضحة في الرسوم التوضيحية. يتم التركيز على فهم كيفية تقسيم القطع المستقيمة والعلاقة بين الأجزاء والكل.
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اختر إجابة واحدة لكل سؤال. عند الاختيار ستظهر النتيجة فورًا: الأخضر صحيح، والأحمر خطأ، وسيظهر تفسير الإجابة مباشرة إن كان متوفرًا. وبعد آخر سؤال ستظهر الدرجة النهائية تلقائيًا.
Find the value of x if Q is between P and R, PQ = 6x + 20, QR = 2(x + 6), and $\overline{PQ} \cong \overline{QR}$.
Explanation
Since the segments are congruent, their lengths are equal: 6x + 20 = 2(x + 6). Expanding gives 6x + 20 = 2x + 12. Subtracting 2x and 20 from both sides results in 4x = -8, so x = -2.
Question 2
Points: 1
Find the measure of segment $\overline{MO}$ based on the given diagram.
Explanation
The tick marks indicate that MN = NO. Given MN = 4.6 cm, then NO = 4.6 cm. Therefore, MO = MN + NO = 4.6 + 4.6 = 9.2 cm.
Question 3
Points: 1
Find the measure of segment $\overline{WY}$ based on the given diagram.
Explanation
The total length WX is 8.8 mm. The tick marks show that WY = YX. Thus, WY is half of WX: 8.8 / 2 = 4.4 mm.
Question 4
Points: 1
Find the length of $\overline{FG}$ in the diagram below.
Explanation
The entire segment FK is 16.8 cm long and is divided into 4 congruent parts (FG, GH, HJ, and JK). Therefore, FG = 16.8 / 4 = 4.2 cm.
Question 5
Points: 1
Find the length of segment $\overline{QT}$ based on the diagram.
Explanation
The total segment PT is 14.4 inches, divided into 4 equal parts (PQ, QR, RS, ST). Each part is 14.4 / 4 = 3.6 inches. Segment QT consists of 3 of these parts (QR + RS + ST), so QT = 3 × 3.6 = 10.8 inches.
Question 6
Points: 1
Find the length of $\overline{DE}$ using the information in the diagram.
Explanation
The tick marks show CD = DE, so 2x + 7 = 4(x - 3). Solving the equation: $2x + 7 = 4x - 12 \Rightarrow 19 = 2x \Rightarrow x = 9.5$. Thus, DE = 4(9.5 - 3) = 4(6.5) = 26.
Question 7
Points: 1
Find the length of segment $\overline{UX}$ based on the diagram.
Explanation
Tick marks show UV = VW = WX. Setting 3x + 1 = 4x - 6 gives x = 7. The length of one segment is 3(7) + 1 = 22. Since there are 3 congruent segments, UX = 22 × 3 = 66 units.
Question 8
Points: 1
Find the value of x and BC if B is between A and C, AC = 4x - 12, AB = x, and BC = 2x + 3.
Explanation
By the segment addition postulate, AB + BC = AC, so x + (2x + 3) = 4x - 12. Simplifying gives 3x + 3 = 4x - 12, so x = 15. Then BC = 2(15) + 3 = 33.
Question 9
Points: 1
Find the length of $\overline{UW}$ if W is between U and V, UV = 16.8 centimeters, and VW = 7.9 centimeters.
Explanation
Since W is between U and V, UW + VW = UV. Therefore, UW = UV - VW = 16.8 - 7.9 = 8.9 cm.
Question 10
Points: 1
Find the value of x if RS = 24 centimeters.
Explanation
In the diagram, RT + TS = RS. Substituting the values gives (6x - 4) + 10 = 24. Simplifying gives 6x + 6 = 24, so 6x = 18 and x = 3.
Question 11
Points: 1
Find the length of $\overline{LO}$ if M is between L and O, LM = 7x - 9, MO = 14 inches, and LO = 10x - 7.
Explanation
Using LM + MO = LO, we set up (7x - 9) + 14 = 10x - 7. This simplifies to 7x + 5 = 10x - 7, so 3x = 12 and x = 4. The length of LO is 10(4) - 7 = 33 inches.
Question 12
Points: 1
Find the value of x if $\overline{PQ} \cong \overline{RS}$, PQ = 9x - 7, and RS = 29.
Explanation
Congruent segments have equal lengths, so 9x - 7 = 29. Adding 7 to both sides gives 9x = 36, so x = 4.
Question 13
Points: 1
Find the value of the variable and YZ if Y is between X and Z, XY = 11, YZ = 4c, XZ = 83.
Explanation
By segment addition, 11 + 4c = 83. Subtracting 11 gives 4c = 72, so c = 18. Thus, YZ = 4(18) = 72.
Question 14
Points: 1
Find the value of the variable and YZ if Y is between X and Z, XY = 6b, YZ = 8b, XZ = 175.
Explanation
Summing the parts: 6b + 8b = 175, so 14b = 175. Dividing by 14 gives b = 12.5. Therefore, YZ = 8(12.5) = 100.
Question 15
Points: 1
Find the value of the variable and YZ if Y is between X and Z, XY = 7a, YZ = 5a, XZ = 6a + 24.
Explanation
We have 7a + 5a = 6a + 24. Combining like terms gives 12a = 6a + 24. Subtracting 6a gives 6a = 24, so a = 4. Thus YZ = 5(4) = 20.
Question 16
Points: 1
Find the value of the variable and YZ if Y is between X and Z, XY = 5.5, YZ = 2c, XZ = 8.9.
Explanation
We use 5.5 + 2c = 8.9. Subtracting 5.5 gives 2c = 3.4, so c = 1.7. Then YZ = 2(1.7) = 3.4.
Question 17
Points: 1
Find the value of the variable and YZ if Y is between X and Z, XY = 5n, YZ = 2n, XZ = 91.
Explanation
Equation: $5n + 2n = 91 \Rightarrow 7n = 91$. Dividing by 7 gives n = 13. YZ = 2(13) = 26.
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