المثال 2 - كتاب الرياضيات - الصف 10 - الفصل 2 - المملكة العربية السعودية

المثال 2: حدد ما إذا كان ZW || VY أم لا، وبرر إجابتك في كل من السؤالين الآتيين:

12) ZX = 18, ZV = 6, WX = 24, YX = 16

13) WX = 31, YX = 21, ZX = 4ZV

المثال 3: في KLM ، إذا كانت JP , JH , PH قطعًا منصفة ، فأوجد قيمة x في كل من السؤالين الآتيين:

14) (The question refers to the diagram with angles 44°, 76°, x°)

15) (The question refers to the diagram with lengths x and 2.7)

المثال 4: خرائط:

16) المسافة من مدخل الحديقة إلى طريق المشاة على امتداد الطريق المرصوف 880m . إذا كان طريق المشاة يوازي الطريق الترابي، فأوجد المسافة من مدخل الحديقة إلى طريق المشاة على امتداد منطقة الأشجار.

المثال 5: جبر: أوجد قيمة كل من y , x في السؤالين الآتيين:

17) (The question refers to the diagram with expressions 20 - 5x, 2x + 6, -y, 3/5y + 2)

18) (The question refers to the diagram with expressions 1/3x + 2, 2/3x - 4, 5y, 7/3y + 8)

برهان: اكتب برهانًا حرًّا لكل مما يأتي:

19) النتيجة 6.1

20) النتيجة 6.2

برهان: اكتب برهانًا ذا عمودين للنظريتين الآتيتين:

21) النظرية 6.5

22) النظرية 6.6

23) النظرية 6.7

استعمل QRS للإجابة عن السؤالين الآتيين:

24) إذا كان: 6 = PT ، 4 = TR ، 8 = ST ، فأوجد QR .

25) إذا كان: 12 = QR ، 6 = PT ، 4 = SP ، فأوجد SQ .

26) إذا كان: 2 - CE = t ، 1 + EB = t ، 10 = CA ، 2 = CD ، فأوجد قيمة كل من t , CE .

27) إذا كان: 2 : LK = 4 ، 3 = MP ، 6 = PQ ، 2 = KJ ، 6 = RS ، 2 = LP ، فأوجد قيمة كل من ML, QR, QK, JH .

وزارة التعليم Ministry of Education 2023 - 1447

96 الفصل 6 التشابه

--- SECTION: المثال 2 --- المثال 2: حدد ما إذا كان ZW || VY أم لا، وبرر إجابتك في كل من السؤالين الآتيين: --- SECTION: 12 --- 12) ZX = 18, ZV = 6, WX = 24, YX = 16 --- SECTION: 13 --- 13) WX = 31, YX = 21, ZX = 4ZV --- SECTION: المثال 3 --- المثال 3: في KLM ، إذا كانت JP , JH , PH قطعًا منصفة ، فأوجد قيمة x في كل من السؤالين الآتيين: --- SECTION: 14 --- 14) (The question refers to the diagram with angles 44°, 76°, x°) --- SECTION: 15 --- 15) (The question refers to the diagram with lengths x and 2.7) --- SECTION: المثال 4 --- المثال 4: خرائط: --- SECTION: 16 --- 16) المسافة من مدخل الحديقة إلى طريق المشاة على امتداد الطريق المرصوف 880m . إذا كان طريق المشاة يوازي الطريق الترابي، فأوجد المسافة من مدخل الحديقة إلى طريق المشاة على امتداد منطقة الأشجار. --- SECTION: المثال 5 --- المثال 5: جبر: أوجد قيمة كل من y , x في السؤالين الآتيين: --- SECTION: 17 --- 17) (The question refers to the diagram with expressions 20 - 5x, 2x + 6, -y, 3/5y + 2) --- SECTION: 18 --- 18) (The question refers to the diagram with expressions 1/3x + 2, 2/3x - 4, 5y, 7/3y + 8) --- SECTION: برهان --- برهان: اكتب برهانًا حرًّا لكل مما يأتي: --- SECTION: 19 --- 19) النتيجة 6.1 --- SECTION: 20 --- 20) النتيجة 6.2 --- SECTION: برهان --- برهان: اكتب برهانًا ذا عمودين للنظريتين الآتيتين: --- SECTION: 21 --- 21) النظرية 6.5 --- SECTION: 22 --- 22) النظرية 6.6 --- SECTION: 23 --- 23) النظرية 6.7 استعمل QRS للإجابة عن السؤالين الآتيين: --- SECTION: 24 --- 24) إذا كان: 6 = PT ، 4 = TR ، 8 = ST ، فأوجد QR . --- SECTION: 25 --- 25) إذا كان: 12 = QR ، 6 = PT ، 4 = SP ، فأوجد SQ . --- SECTION: 26 --- 26) إذا كان: 2 - CE = t ، 1 + EB = t ، 10 = CA ، 2 = CD ، فأوجد قيمة كل من t , CE . --- SECTION: 27 --- 27) إذا كان: 2 : LK = 4 ، 3 = MP ، 6 = PQ ، 2 = KJ ، 6 = RS ، 2 = LP ، فأوجد قيمة كل من ML, QR, QK, JH . وزارة التعليم Ministry of Education 2023 - 1447 96 الفصل 6 التشابه --- VISUAL CONTEXT --- **DIAGRAM**: Untitled Description: A large triangle ZWX with a line segment VY drawn such that V is on side ZX and Y is on side WX. The line segment VY appears parallel to ZW, but this needs to be determined. The vertices are labeled Z, W, X, V, Y. Key Values: Z, W, X, V, Y Context: Used to determine if a line segment is parallel to a side of a triangle based on segment ratios, related to the Triangle Proportionality Theorem or its converse. **DIAGRAM**: Untitled Description: A triangle KLM. A point P is on side KM, J on KL, and H on ML. Segments PH, JH, JP are drawn. Angle KPH is labeled 44°. Angle HJL is labeled 76°. Angle JPH is labeled x°. The problem statement indicates that JP, JH, PH are 'bisecting segments', which might imply properties related to angle bisectors or medians, but the diagram specifically shows angles formed by these segments within the triangle. Key Values: K, L, M, P, J, H, 44°, 76°, x° Context: Used to find an unknown angle x in a geometric figure, likely using properties of triangles and angle relationships. **DIAGRAM**: Untitled Description: A triangle KLM. A line segment JH is drawn inside the triangle, with J on KL and H on ML. JH is marked with double arrows, indicating it is parallel to KM. The length of JH is labeled 'x'. The length of KM is labeled '2.7'. The problem statement indicates that JP, JH, PH are 'bisecting segments', which in this context, given the parallel lines, might refer to JH being a midsegment or a segment that divides the sides proportionally. Key Values: K, L, M, J, H, P, x, 2.7 Context: Used to find an unknown length x in a geometric figure, likely using properties of similar triangles formed by a line parallel to one side (Triangle Proportionality Theorem or similar triangles). **DIAGRAM**: Untitled Description: A diagram representing a map. It shows a 'مدخل حديقة عامة' (public park entrance) at the top. A 'طريق مرصوف' (paved road) runs vertically on the left, with a length of '1408 m' indicated. A 'طريق مشاة' (pedestrian path) runs parallel to a 'طريق ترابي' (dirt road) further to the right. A 'منطقة أشجار' (tree area) is shown on the right side, with a length of '1760 m' indicated along its edge. The pedestrian path is shown as a segment between the paved road and the tree area. The problem states the pedestrian path is parallel to the dirt road. The distance from the park entrance to the pedestrian path along the paved road is 880m. The diagram shows a triangular shape formed by the park entrance and the roads, labeled 'مثلثية الشكل' (triangular shape). The problem asks to find the distance from the park entrance to the pedestrian path along the tree area. Key Values: مدخل حديقة عامة, طريق مرصوف, طريق مشاة, طريق ترابي, منطقة أشجار, 1408 m, 1760 m, 880 m Context: Used to solve a real-world problem involving distances and parallel lines, likely applying the Triangle Proportionality Theorem or similar triangles to find an unknown distance. **DIAGRAM**: Untitled Description: A diagram showing three parallel horizontal lines. Two transversal lines intersect these parallel lines. The segments formed on the left transversal are labeled '20 - 5x' (top) and '2x + 6' (bottom). The segments formed on the right transversal are labeled '-y' (top) and '3/5y + 2' (bottom). The parallel lines are indicated by arrows. The problem asks to find x and y. Key Values: 20 - 5x, 2x + 6, -y, 3/5y + 2 Context: Used to solve for unknown variables x and y using the properties of parallel lines and transversals, specifically the Parallel Proportionality Theorem (or Three Parallel Lines Theorem) which states that parallel lines cut transversals proportionally. **DIAGRAM**: Untitled Description: A triangle with a horizontal line segment inside, parallel to the base. The parallel lines are indicated by double arrows. The top segment of the left side of the triangle is labeled '1/3x + 2'. The bottom segment of the left side is labeled '2/3x - 4'. The top segment of the right side of the triangle is labeled '5y'. The bottom segment of the right side is labeled '7/3y + 8'. The problem asks to find x and y. Key Values: 1/3x + 2, 2/3x - 4, 5y, 7/3y + 8 Context: Used to solve for unknown variables x and y using the Triangle Proportionality Theorem, which states that if a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. **DIAGRAM**: Untitled Description: A triangle QRS. A line segment PT is drawn inside the triangle, with P on side QR and T on side RS. The segment PT is marked with single arrows, indicating it is parallel to side QS. The vertices are labeled Q, R, S, P, T. Key Values: Q, R, S, P, T Context: Used to solve problems involving similar triangles formed by a line parallel to one side, applying the Triangle Proportionality Theorem or properties of similar triangles. **DIAGRAM**: Untitled Description: A triangle ABC. A line segment DE is drawn inside the triangle, with D on side AC and E on side BC. The segment DE is marked with single arrows, indicating it is parallel to side AB. The vertices are labeled A, B, C, D, E. Key Values: A, B, C, D, E Context: Used to solve problems involving similar triangles formed by a line parallel to one side, applying the Triangle Proportionality Theorem or properties of similar triangles. **DIAGRAM**: Untitled Description: A large triangle HSM. Points J, K, L are on side HM. Points R, Q, P are on side SM. Segments JR, QK, LP are drawn. These segments are marked with single arrows, indicating they are parallel to each other and to the base SM. The problem statement implies that these segments are parallel. The vertices are labeled H, S, M, J, K, L, R, Q, P. Key Values: H, S, M, J, K, L, R, Q, P Context: Used to solve problems involving parallel lines cutting transversals proportionally, or similar triangles formed by multiple parallel lines.