Question : In a triangle PQR, S, and T are the points on PQ and PR, respectively, such that ST || QR and $\frac{\text{PS}}{\text{SQ}} = \frac{3}{5}$ and PR = 6 cm. Then PT is:
Option 1: 2 cm
Option 2: 2.25 cm
Option 3: 3.5 cm
Option 4: 4 cm
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Correct Answer: 2.25 cm
Solution : Given: In $\triangle$PQR, QR || ST $\frac{\text{PS}}{\text{SQ}} = \frac{3}{5}$ Let PS = 3 units, SQ = 5 units So, PQ = 3 + 5 = 8 units PR = 6 cm In ∆ PST and ∆ PQR, ∵ QR | | ST ∵ $\angle$S = $\angle$Q ; $\angle$T = $\angle$R ∴ By AA - similarity, Here, $\triangle$PST $\sim \triangle$PQR $\frac{\text{PS}}{\text{PQ}}=\frac{\text{PT}}{\text{PR}}$ ⇒ $\frac{3}{8}=\frac{\text{PT}}{6}$ $\therefore$ PT = $\frac{18}{8}=2.25$ cm Hence, the correct answer is 2.25 cm.
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Question : A circle is inscribed in $\triangle $PQR touching the sides QR, PR and PQ at the points S, U and T, respectively. PQ = (QR + 5) cm, PQ = (PR + 2) cm. If the perimeter of $\triangle $PQR is 32 cm, then PR is equal to:
Question : S and T are points on the sides PQ and PR, respectively, of $\triangle$PQR such that PS × PR = PQ × PT. If $\angle$Q = 96° and $\angle$PST = $\angle$PRQ + 34°, then $\angle$QPR = ?
Question : Let A, B, and C be the mid-points of sides PQ, QR, and PR, respectively, of PQR. If the area of $\triangle$ PQR is 32 cm2, then find the area of $\triangle$ ABC.
Question : In $\triangle P Q R, \angle P=90^{\circ}. {S}$ and ${T}$ are the mid points of sides ${PR}$ and ${PQ}$, respectively. What is the value of $\frac{\text{RQ}^2}{(\text{QS}^2 + \text{RT}^2)}$?
Question : $\triangle$ABC is similar to $\triangle$PQR and AB : PQ = 2 : 3. AD is the median to the side BC in $\triangle$ABC and PS is the median to the side QR in $\triangle$PQR. What is the value of $(\frac{BD}{QS})^2$?
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