. PS bisects angle QPR and PS is perpendicular to QR. If PQ = 2x
units, PR = (3y + 8) units, QS = x units and
SR = 2y units. Find the values of x and y.
Answers
Given:
PS bisects angle QPR and PS is perpendicular to QR. If PQ = 2x units, PR = (3y + 8) units, QS = x units and SR = 2y units. Find the values of x and y.
To find:
The values of x and y
Solution:
PS bisects ∠QPR . . . (given)
∴ ∠QPS = ∠RPS . . . (1)
PS is perpendicular to QR
∴ ∠PSQ = ∠PSR = 90° . . . (2)
In Δ PQS and Δ PRS, we get
∠QPS = ∠RPS . . . [from (1)]
PS = PS . . . [common side to both the triangles]
∠PSQ = ∠PSR = 90° . . . [from (2)]
∴ Δ PQS ≅ Δ PRS . . . [by ASA congruency]
We know that,
If two triangles are congruent to each other, then all 3 sides will have equal length and all 3 angles will have equal measurements.
∵ Δ PQS ≅ Δ PRS, so, we get:
QS = SR
on substituting the given values
⇒ x = 2y . . . (3)
and
PQ = PR
on substituting the given values
⇒ 2x = 3y + 8
on substituting from (3)
⇒ 2 × 2y = 3y + 8
⇒ 4y = 3y + 8
⇒ 4y - 3y = 8
⇒ y = 8
On substituting the value of y = 8 in (3), we get
x = 2y = 2 × 8 = 16
Thus, the values of x and y are:
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Answer:
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