In the given figure, PQRS is a parallelogram and SPQ = 60°. The bisector PX and QX of angles P and Q respectively, meet at X on RS. If PQ = 18 cm then RQ = ?
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In the given figure, PQRS is a parallelogram and SPQ = 60°. The bisector PX and QX of angles P and Q respectively, meet at X on RS. If PQ = 18 cm then RQ = ?
pls answer ill mark as brainliest
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This is a geometry problem involving parallelograms and angle bisectors. I can try to help you solve it using some facts and formulas. Here are the steps:
- First, we need to find the measure of ∠P and ∠Q. Since PQRS is a parallelogram, we know that opposite angles are equal and adjacent angles are supplementary. Therefore, we have ∠P = ∠R and ∠Q = ∠S. Also, ∠P + ∠Q = 180°.
- Since SPQ = 60°, we can use the fact that the sum of the interior angles of a triangle is 180° to find ∠S. We have ∠S + 60° + ∠R = 180°. Substituting ∠R with ∠P, we get ∠S + 60° + ∠P = 180°. Simplifying, we get ∠S = 120° - ∠P.
- Now, we can use the equation ∠P + ∠Q = 180° and substitute ∠Q with ∠S to find ∠P. We have ∠P + (120° - ∠P) = 180°. Solving for ∠P, we get ∠P = 30°. Therefore, ∠Q = 180° - 30° = 150°.
- Next, we need to find the length of PX and QX. Since PX and QX are the bisectors of ∠P and ∠Q respectively, we can use the angle bisector theorem to find the ratio of PX to XR and QX to XS. The angle bisector theorem states that if a point D lies on the side BC of a triangle ABC such that AD bisects ∠A, then BD / DC = AB / AC¹.
- Applying the angle bisector theorem to triangles PQX and QXR, we get PX / XR = PQ / QR and QX / XS = QR / QS respectively. Since PQRS is a parallelogram, we know that opposite sides are equal, so PQ = SR and QR = PS. Therefore, we can rewrite the ratios as PX / XR = SR / QR and QX / XS = QR / SR.
- Now, we can use the fact that X is the point of intersection of PX and QX to find a relation between PX and QX. We have PX + XR = QX + XS. Substituting XR with PX ⋅ QR / SR and XS with QX ⋅ SR / QR, we get PX + (PX ⋅ QR / SR) = QX + (QX ⋅ SR / QR). Simplifying, we get PX ⋅ (SR + QR) = QX ⋅ (SR + QR).
- Since SR + QR is not zero, we can divide both sides by it and get PX = QX. This means that X is the midpoint of RS.
- Finally, we can use the Pythagorean theorem to find RQ. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides². Applying this to triangle RQX, we get RQ^2 = RX^2 + QX^2.
- Since X is the midpoint of RS, we have RX = RS / 2. Since PQRS is a parallelogram, we have RS = PQ. Therefore, RX = PQ / 2. Substituting PQ with 18 cm, we get RX = 9 cm.
- Since PX = QX, we have QX = PX. Using the angle bisector theorem again, we have PX / XR = PQ / QR or PX ⋅ QR = PQ ⋅ XR. Substituting PQ with 18 cm and XR with RX or 9 cm, we get PX ⋅ QR = 18 ⋅ 9 or PX ⋅ QR = 162 cm^2.
- Solving for QR, we get QR = 162 / PX. Substituting this into RQ^2 = RX^2 + QX^2, we get RQ^2 = (9)^2 + (162 / PX)^2.
- Simplifying, we get RQ^2 = 81 + (26244 / PX^2). To find RQ, we need to take the square root of both sides. We get RQ = sqrt(81 + (26244 / PX^2)).
- This is the final answer in terms of PX. To find the exact value of RQ, we need to know the value of PX which depends on the length of SR or PQ.
I hope this helps you understand how to solve this problem. If you have any questions or feedback, please let me know