PQRS is a rectangle. The perpendicular ST FROM S ON PR DIVIDES ANGLES IN RATIO 2:3 .FIND ANGLE TPQ.
To find angle TPQ, we need to use the information given in the problem. Let's break it down step-by-step:
1. We know that PQRS is a rectangle. In a rectangle, opposite angles are equal. Therefore, angle PSR is equal to angle QSP.
2. The problem states that the perpendicular ST from S on PR divides angles in the ratio 2:3.
3. Let's label the angle PSR as x. Since angle QSP is equal to angle PSR, we can also label angle QSP as x.
4. According to the ratio given in the problem, the angle STQ can be represented as 2x, and the angle STP can be represented as 3x.
5. Now, let's look at triangle STQ. The sum of the angles in a triangle is 180 degrees. So, we can write an equation: 2x + 3x + angle TPQ = 180.
6. Simplifying the equation, we have 5x + angle TPQ = 180.
7. We can substitute the values of x and angle TPQ into the equation using the information we have so far. x = angle PSR = angle QSP, and we need to find angle TPQ.
8. To find angle TPQ, we need to solve for x. Subtracting angle TPQ from both sides of the equation, we have 5x = 180 - angle TPQ.
9. Now, let's assume that angle TPQ is y. Substituting this into the equation, we have 5x = 180 - y.
10. To find the value of x, we need to solve the equation using other information given in the problem. Unfortunately, we don't have enough information to find the exact values of x or y.
To find the angle TPQ, we can use the information given about the division of angles in the rectangle PQRS.
Let's analyze the problem step by step:
1. Draw the diagram: Begin by drawing the rectangle PQRS.
P ______ Q
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| |
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S ______ R
2. Identify the given information: We are told that the perpendicular ST divides the angles in the rectangle PQRS in a ratio of 2:3.
3. Label the diagram: Label the angle at point P as TPQ and let the point of intersection of ST and PR be labeled as O.
P ______ Q
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| O |
S ______ R
4. Use the given ratio: Since the perpendicular ST divides the angles in a 2:3 ratio, we can determine that angle RSO is twice as large as angle OSP.
5. Determine the angle relation: Since angles TPQ and RSO are vertically opposite angles (opposite each other at the intersection of PR and ST), they are equal.
6. Calculate the angle TPQ: Given that angle TPQ is equal to angle RSO (from step 5), we can now determine the angle TPQ using the 2:3 ratio. Let's represent angle TPQ as x:
2x + 3x = 180 degrees (sum of the two angles is equal to 180 degrees because they form a straight line)
5x = 180 degrees (combine like terms)
x = 180 degrees / 5 (divide both sides by 5)
x = 36 degrees
Therefore, angle TPQ is equal to 36 degrees.
Note that ∆STP is similar to ∆RTS
and angle SRT = angle TPQ (alternate interior angles)