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In a APQR, PQ = PR = 11cm and S is a point on QR such that PS = 10cm. If the lengths of QB and SR, when expressed in cm, are integers, then find the length of QR.
- a)9cm
- b)4.58cm
- c)11cm
- d)Cannot be determined
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
In a APQR, PQ = PR = 11cm and S is a point on QR such that PS = 10cm. ...
To find the length of QR, we need to first determine the lengths of QB and SR.
From the given information, PQ = PR = 11 cm, and PS = 10 cm. Let's denote QB as x and SR as y.
Using the triangle inequality theorem, we know that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we can set up two inequalities:
1) QB + PS > QS
2) SR + PS > PQ
Simplifying these inequalities, we get:
1) x + 10 > QS
2) y + 10 > 11
From the first inequality, we can deduce that x > QS - 10. Since QS is a part of QR, the maximum value of QS can be QR itself. So, x > QR - 10.
From the second inequality, we have y > 1.
Now, let's consider the extremes. The maximum value of x is QR - 10, and the minimum value of y is 1.
Therefore, the maximum value of x + y is (QR - 10) + 1 = QR - 9.
Since QB and SR are integers, x + y must be an integer. Therefore, QR - 9 must also be an integer.
The only option that satisfies this condition is option B, where QR is 4.58 cm.
From the given information, PQ = PR = 11 cm, and PS = 10 cm. Let's denote QB as x and SR as y.
Using the triangle inequality theorem, we know that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we can set up two inequalities:
1) QB + PS > QS
2) SR + PS > PQ
Simplifying these inequalities, we get:
1) x + 10 > QS
2) y + 10 > 11
From the first inequality, we can deduce that x > QS - 10. Since QS is a part of QR, the maximum value of QS can be QR itself. So, x > QR - 10.
From the second inequality, we have y > 1.
Now, let's consider the extremes. The maximum value of x is QR - 10, and the minimum value of y is 1.
Therefore, the maximum value of x + y is (QR - 10) + 1 = QR - 9.
Since QB and SR are integers, x + y must be an integer. Therefore, QR - 9 must also be an integer.
The only option that satisfies this condition is option B, where QR is 4.58 cm.
Community Answer
In a APQR, PQ = PR = 11cm and S is a point on QR such that PS = 10cm. ...
By using Pythagoras theorem
hypotenuse=Base×height
pq²=qs²+ps²
11²=qs²+10²
121=qs²+100
qs²=121-100
qs²=21
qs=√21
qs=4.5825
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In a APQR, PQ = PR = 11cm and S is a point on QR such that PS = 10cm. If the lengths of QB and SR, when expressed in cm, are integers, then find the length of QR.a)9cmb)4.58cmc)11cmd)Cannot be determinedCorrect answer is option 'B'. Can you explain this answer?
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