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In figure, PS is the bisector of angle QPR and PT is perpendicular to QR. Show that angle TPS=1/2 (Angle Q-Angle R)?
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In figure, PS is the bisector of angle QPR and PT is perpendicular to ...
Given; PS is bisector of angle QPR
PT is perpendicular to QR
To Prove; angle TPS = 1/2 (angle Q - angle R)
Solutio; We know that angle QPS = 1/2 angle P
(1) angle Q + angle QPT = 90 degree
angle QPT = 90 degree -angle Q (2) On putting equations (1) and (2)
angle TPS = angle QPS - angle QPT
= 1/2 angle P - ( 90 - angle Q )
= 1/2 angle P - 90 + angle Q
= 1/2 angle P - 1/2 (angle Q + angle P + angle R) + angle Q
= 1/2 angle P - 1/2 angle Q - 1/2 angle P - 1/2 angle R + angle Q
= -1/2 angle Q - 1/2 angle R + angle Q (+ve and -ve 1/2 angle P got cancled )
by takig LCM in angle Q we get
-1/2 angle Q + 2/2 angle Q - 1/2 angle R
(-1/2 + 2/2) angle Q - 1/2 angle R
= 1/2 angle Q - 1/2 angle R
angle TPS = 1/2 ( angle Q -1/2 angle R ) (Hence proved)
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Most Upvoted Answer
In figure, PS is the bisector of angle QPR and PT is perpendicular to ...
To show that angle TPS is equal to half of angle Q minus angle R, we can use the properties of angles formed by the intersection of lines and angles in a triangle.
Given:
- PS is the bisector of angle QPR
- PT is perpendicular to QR
To Prove:
- Angle TPS = 1/2 (Angle Q - Angle R)
Proof:
1. Angle Bisector Property:
Since PS is the bisector of angle QPR, it divides the angle into two equal angles. Therefore, angle TPS is equal to angle TPR.
2. Angle Sum Property:
In triangle PQR, the sum of the angles is equal to 180 degrees. Therefore, we can write the equation:
Angle Q + Angle R + Angle P = 180 degrees
3. Right Triangle Property:
Since PT is perpendicular to QR, angle PTR is a right angle.
4. Angle Sum Property in a Triangle:
In triangle PTR, the sum of the angles is equal to 180 degrees. Therefore, we can write the equation:
Angle P + Angle T + Angle R = 180 degrees
5. Substitution:
We can substitute the value of Angle P from equation 2 into equation 4:
(Angle Q + Angle R) + Angle T + Angle R = 180 degrees
2 * Angle R + Angle Q + Angle T = 180 degrees
6. Rearranging the Equation:
We can rearrange the equation by moving Angle Q to the other side:
2 * Angle R + Angle T = 180 degrees - Angle Q
7. Angle TPS and Angle TPR:
From step 1, we know that angle TPS is equal to angle TPR.
8. Substitution:
We can substitute angle TPR with angle TPS in the equation from step 6:
2 * Angle R + Angle TPS = 180 degrees - Angle Q
9. Angle TPS = 1/2 (Angle Q - Angle R):
We can further rearrange the equation:
Angle TPS = (180 degrees - Angle Q) - 2 * Angle R
Angle TPS = 180 degrees - Angle Q - 2 * Angle R
Angle TPS = 180 degrees - Angle Q + (-2) * Angle R
Angle TPS = 180 degrees - Angle Q + (-2) * (Angle R)
Angle TPS = 180 degrees - Angle Q + (-2) * (-1) * (Angle R)
Angle TPS = 180 degrees - Angle Q + 2 * Angle R
Angle TPS = 180 degrees - (Angle Q - 2 * Angle R)
Angle TPS = 180 degrees - (Angle Q - Angle R) + Angle R
Angle TPS = 180 degrees - 1 * (Angle Q - Angle R) + Angle R
Angle TPS = 180 degrees - 1 * (Angle Q - Angle R) + 1 * Angle R
Angle TPS = 180 degrees - 1 * (Angle Q - Angle R) + 1 * (Angle R)
Angle TPS = 180 degrees - 1 * (Angle Q - Angle R)
Given:
- PS is the bisector of angle QPR
- PT is perpendicular to QR
To Prove:
- Angle TPS = 1/2 (Angle Q - Angle R)
Proof:
1. Angle Bisector Property:
Since PS is the bisector of angle QPR, it divides the angle into two equal angles. Therefore, angle TPS is equal to angle TPR.
2. Angle Sum Property:
In triangle PQR, the sum of the angles is equal to 180 degrees. Therefore, we can write the equation:
Angle Q + Angle R + Angle P = 180 degrees
3. Right Triangle Property:
Since PT is perpendicular to QR, angle PTR is a right angle.
4. Angle Sum Property in a Triangle:
In triangle PTR, the sum of the angles is equal to 180 degrees. Therefore, we can write the equation:
Angle P + Angle T + Angle R = 180 degrees
5. Substitution:
We can substitute the value of Angle P from equation 2 into equation 4:
(Angle Q + Angle R) + Angle T + Angle R = 180 degrees
2 * Angle R + Angle Q + Angle T = 180 degrees
6. Rearranging the Equation:
We can rearrange the equation by moving Angle Q to the other side:
2 * Angle R + Angle T = 180 degrees - Angle Q
7. Angle TPS and Angle TPR:
From step 1, we know that angle TPS is equal to angle TPR.
8. Substitution:
We can substitute angle TPR with angle TPS in the equation from step 6:
2 * Angle R + Angle TPS = 180 degrees - Angle Q
9. Angle TPS = 1/2 (Angle Q - Angle R):
We can further rearrange the equation:
Angle TPS = (180 degrees - Angle Q) - 2 * Angle R
Angle TPS = 180 degrees - Angle Q - 2 * Angle R
Angle TPS = 180 degrees - Angle Q + (-2) * Angle R
Angle TPS = 180 degrees - Angle Q + (-2) * (Angle R)
Angle TPS = 180 degrees - Angle Q + (-2) * (-1) * (Angle R)
Angle TPS = 180 degrees - Angle Q + 2 * Angle R
Angle TPS = 180 degrees - (Angle Q - 2 * Angle R)
Angle TPS = 180 degrees - (Angle Q - Angle R) + Angle R
Angle TPS = 180 degrees - 1 * (Angle Q - Angle R) + Angle R
Angle TPS = 180 degrees - 1 * (Angle Q - Angle R) + 1 * Angle R
Angle TPS = 180 degrees - 1 * (Angle Q - Angle R) + 1 * (Angle R)
Angle TPS = 180 degrees - 1 * (Angle Q - Angle R)
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In figure, PS is the bisector of angle QPR and PT is perpendicular to QR. Show that angle TPS=1/2 (Angle Q-Angle R)?
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In figure, PS is the bisector of angle QPR and PT is perpendicular to QR. Show that angle TPS=1/2 (Angle Q-Angle R)? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In figure, PS is the bisector of angle QPR and PT is perpendicular to QR. Show that angle TPS=1/2 (Angle Q-Angle R)? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In figure, PS is the bisector of angle QPR and PT is perpendicular to QR. Show that angle TPS=1/2 (Angle Q-Angle R)?.
In figure, PS is the bisector of angle QPR and PT is perpendicular to QR. Show that angle TPS=1/2 (Angle Q-Angle R)? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In figure, PS is the bisector of angle QPR and PT is perpendicular to QR. Show that angle TPS=1/2 (Angle Q-Angle R)? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In figure, PS is the bisector of angle QPR and PT is perpendicular to QR. Show that angle TPS=1/2 (Angle Q-Angle R)?.
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