Advertisements
Advertisements
प्रश्न
In ∆PQR, point S is the midpoint of side QR. If PQ = 11, PR = 17, PS = 13, find QR.
Advertisements
उत्तर
In ∆PQR, point S is the midpoint of side QR.
\[{PQ}^2 + {PR}^2 = 2 {PS}^2 + 2 {QS}^2\] .......…[Apollonius theorem]
\[ \Rightarrow {11}^2 + {17}^2 = 2 \left( 13 \right)^2 + 2 {QS}^2 \]
\[ \Rightarrow 121 + 289 = 2\left( 169 \right) + 2 {QS}^2 \]
\[ \Rightarrow 410 = 338 + 2 {QS}^2 \]
\[ \Rightarrow 2 {QS}^2 = 410 - 338\]
\[ \Rightarrow 2 {QS}^2 = 72\]
\[ \Rightarrow {QS}^2 = 36\]
\[ \Rightarrow QS = 6\]
\[ \therefore QR = 2 \times QS\]
\[ = 2 \times 6\]
\[ = 12\]
Hence, QR = 12.
APPEARS IN
संबंधित प्रश्न
The points A(4, 7), B(p, 3) and C(7, 3) are the vertices of a right traingle ,right-angled at B. Find the values of p.
In the following figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that
(i) OA2 + OB2 + OC2 − OD2 − OE2 − OF2 = AF2 + BD2 + CE2
(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2
Tick the correct answer and justify: In ΔABC, AB = `6sqrt3` cm, AC = 12 cm and BC = 6 cm.
The angle B is:
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is
(A)\[7 + \sqrt{5}\]
(B) 5
(C) 10
(D) 12
Identify, with reason, if the following is a Pythagorean triplet.
(11, 60, 61)
Find the length diagonal of a rectangle whose length is 35 cm and breadth is 12 cm.
In the given figure, M is the midpoint of QR. ∠PRQ = 90°. Prove that, PQ2 = 4PM2 – 3PR2.
Pranali and Prasad started walking to the East and to the North respectively, from the same point and at the same speed. After 2 hours distance between them was \[15\sqrt{2}\]
km. Find their speed per hour.
In the figure: ∠PSQ = 90o, PQ = 10 cm, QS = 6 cm and RQ = 9 cm. Calculate the length of PR.
In triangle ABC, angle A = 90o, CA = AB and D is the point on AB produced.
Prove that DC2 - BD2 = 2AB.AD.
In the given figure, BL and CM are medians of a ∆ABC right-angled at A. Prove that 4 (BL2 + CM2) = 5 BC2.
In ∆ ABC, AD ⊥ BC.
Prove that AC2 = AB2 +BC2 − 2BC x BD
A ladder 15m long reaches a window which is 9m above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to other side of the street to reach a window 12m high. Find the width of the street.
In a triangle ABC right angled at C, P and Q are points of sides CA and CB respectively, which divide these sides the ratio 2 : 1.
Prove that : 9(AQ2 + BP2) = 13AB2
PQR is an isosceles triangle with PQ = PR = 10 cm and QR = 12 cm. Find the length of the perpendicular from P to QR.
In a right angled triangle, the hypotenuse is the greatest side
Find the unknown side in the following triangles
Find the distance between the helicopter and the ship
The top of a broken tree touches the ground at a distance of 12 m from its base. If the tree is broken at a height of 5 m from the ground then the actual height of the tree is ______.
