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Find the length of side PS in a composite isosceles triangle figure
Mathematics
Grade 10 of Junior High School
Question Content
In the given figure, in triangle PQS, ∠P = 40°, ∠PQS = 40°; in triangle QRS, QR = 8 cm, RS = 9 cm, ∠R = 56°, ∠QSR = 56°. Find the length of side PS (marked as x cm).
Correct Answer
8 cm
Detailed Solution Steps
1
Step 1: Analyze triangle PQS. Since ∠P = 40° and ∠PQS = 40°, triangle PQS is an isosceles triangle with ∠P = ∠PQS. In an isosceles triangle, the sides opposite equal angles are equal, so PS = QS.
2
Step 2: Analyze triangle QRS. Since ∠R = 56° and ∠QSR = 56°, triangle QRS is an isosceles triangle with ∠R = ∠QSR. In an isosceles triangle, the sides opposite equal angles are equal, so QS = QR.
3
Step 3: From step 1 and step 2, we can deduce that PS = QS = QR. Given QR = 8 cm, so PS = x = 8 cm.
Knowledge Points Involved
1
Isosceles Triangle Theorem (Base Angles Theorem)
This theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. It is used to identify equal sides in a triangle when two equal angles are known, and it applies to all triangles in Euclidean geometry.
2
Transitive Property of Equality
This property states that if a = b and b = c, then a = c. In geometry, it is often used to link equal line segments or angles that are both equal to a common third segment or angle, allowing us to establish direct equality between two unrelated elements.
3
Triangle Angle-Side Relationship
This refers to the core relationship between the angles and sides of a triangle: larger angles correspond to longer opposite sides, and equal angles correspond to equal opposite sides. It is the foundational logic for solving side-length problems when angle measures are known, and vice versa.
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