Advertisements
Advertisements
प्रश्न
In a right-angled triangle PQR, right-angled at Q, S and T are points on PQ and QR respectively such as PT = SR = 13 cm, QT = 5 cm and PS = TR. Find the length of PQ and PS.
Advertisements
उत्तर
In ΔPQT, ∠Q = 90°
∴ PT2 = PQ2 + QT2 ....(By Pythagoras Theorem)
⇒ PQ2 = PT2 - QT2
⇒ PQ2 = PT2 - QT2
= 132 - 52
= 169 - 25
= 144
⇒ PQ = 12cm
Now, PS = TR = a (say)
In ΔSQR, ∠Q = 90°
∴ SR2 = QS2 + QR2 ....(By Pythagoras Theorem)
⇒ SR2 = (PQ - PS)2 + (QT + TR)2
⇒ SR2 = (PQ - PS)2 + (QT + PS)2
⇒ SR2 = PQ2 - 2 x PQ x PS + PS2 + QT2 + 2 x QT x PS + PS2
⇒ 132 = 122 - 2 x 12 x a + a2 + 52 + 2 x 5 x a + a2
⇒ 169 - 144 - 24a + a2 + 25 + 10a + a2
⇒ 169 = 169 - 14a + 2a2
⇒ 2a2 = 14a
⇒ a = 7
Hence, PS = 7cm.
APPEARS IN
संबंधित प्रश्न
Prove that the diagonals of a rectangle ABCD, with vertices A(2, -1), B(5, -1), C(5, 6) and D(2, 6), are equal and bisect each other.
Sides of triangle are given below. Determine it is a right triangle or not? In case of a right triangle, write the length of its hypotenuse. 50 cm, 80 cm, 100 cm
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2, prove that ABC is a right triangle.
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals
In the given figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:
`"AC"^2 = "AD"^2 + "BC"."DM" + (("BC")/2)^2`
The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.
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.
(5, 12, 13)
For finding AB and BC with the help of information given in the figure, complete following activity.
AB = BC .......... 
∴ ∠BAC =
∴ AB = BC = × AC
= × `sqrt8`
= × `2sqrt2`
=
A man goes 40 m due north and then 50 m due west. Find his distance from the starting point.
The given figure shows a quadrilateral ABCD in which AD = 13 cm, DC = 12 cm, BC = 3 cm and ∠ABD = ∠BCD = 90o. Calculate the length of AB.
In a quadrilateral ABCD, ∠B = 90° and ∠D = 90°.
Prove that: 2AC2 - AB2 = BC2 + CD2 + DA2
A boy first goes 5 m due north and then 12 m due east. Find the distance between the initial and the final position of the boy.
The foot of a ladder is 6m away from a wall and its top reaches a window 8m above the ground. If the ladder is shifted in such a way that its foot is 8m away from the wall to what height does its tip reach?
The perpendicular PS on the base QR of a ∆PQR intersects QR at S, such that QS = 3 SR. Prove that 2PQ2 = 2PR2 + QR2
In the given figure, ∠T and ∠B are right angles. If the length of AT, BC and AS (in centimeters) are 15, 16, and 17 respectively, then the length of TC (in centimeters) is ______.
In ΔABC, if DE || BC, AD = x, DB = x – 2, AE = x + 2 and EC = x – 1, then value of x is ______.
A 5 m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4 m high. If the foot of the ladder is moved 1.6 m towards the wall, then find the distance by which the top of the ladder would slide upwards on the wall.
In ∆PQR, PD ⊥ QR such that D lies on QR. If PQ = a, PR = b, QD = c and DR = d, prove that (a + b)(a – b) = (c + d)(c – d).
