Question

In a rhombus PQRS if PQ = 7.5 then find QR.
If ZQPS = 75° then find the measure of ZPQR and ZSRQ.​

Answer 1

Question:-

• In a rhombus PQRS if PQ = 7.5 then find QR. If ∠QPS = 75° then find the measure of ∠PQR and ∠SRQ.

Given:-

• PQ = 7.5
• ∠QPS = 75°

To Find:-

• Find QR.
• find the measure of ∠PQR and ∠SRQ.

Solution:-

All the sides of a rhombus are equal.

(i) PQ = 7.5 cm (given)

• PQRS is a rhombus .

∴ QR = PQ

• Sides of a rhombus are congruent.

∴QR = 7.5cm ______________________________

(ii) ∠QPS = 75° (given)

∠QPS + ∠PQR = 180°

• Adjacent angles of a rhombus are supplementry

∴ 75° + ∠PQR = 180°

∴ ∠PQR = 180° - 75°

∴ ∠PQR = 105°

______________________________

(iii) ∠SRQ = ∠OPS (opposite angles of rhombus)

∴ ∠SRQ = 75°

______________________________

Hence,

• ∠QR = 7.5cm
• ∠PQR = 105°
• ∠SRQ = 75°

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Answer 2

Refer to the attachment for the figure.

$\mathcal{Given}$

• rhombus PQRS
• PQ = 7.5cm

$\mathcal{To\:find}$

• QR

$\mathcal{Concept\:applied}$

• all the sides of a rhombus are equal.
• hence, PQ = QR = 7.5cm.

__________________________

$\mathcal{Given}$

• <QPS = 75⁰

$\mathcal{To\:find}$

• < PQR and <SRQ

$\mathcal{Concepts}$

• the adjacent angles of a rhombus sums up to 180⁰
• opposite angles of a rhombus are equal

$\mathcal{Solution}$

$\longrightarrow$ 75⁰+ <QPS = 180⁰

$\longrightarrow$ <QPS = 105⁰

$\longrightarrow$ 75⁰ = <SRQ

__________________________

$\mathcal{Additional\:information}$

• area of a rhombus :-

$\underline{\boxed{\sf area = \dfrac{1}{2}×d_{1}×d_{2}}}$

here,

• $\sf d_{1}$ = diagonal 1 of rhombus
• $\sf d_{2}$ = diagonal 2 of rhombus