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.
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
In figure, ∠B of ∆ABC is an acute angle and AD ⊥ BC, prove that AC2 = AB2 + BC2 – 2BC × BD
ABC is a right-angled triangle, right-angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angle are 5 cm and 12 cm. Find the radius of the circle
In Figure ABD is a triangle right angled at A and AC ⊥ BD. Show that AC2 = BC × DC
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals
D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE2 + BD2 = AB2 + DE2
The perpendicular from A on side BC of a Δ ABC intersects BC at D such that DB = 3CD . Prove that 2AB2 = 2AC2 + BC2.
Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, ho much string does she have out (see Figure)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?
Identify, with reason, if the following is a Pythagorean triplet.
(3, 5, 4)
In right angle ΔABC, if ∠B = 90°, AB = 6, BC = 8, then find AC.
Find the Pythagorean triplet from among the following set of numbers.
9, 40, 41
The sides of the triangle are given below. Find out which one is the right-angled triangle?
1.5, 1.6, 1.7
Find the length of the perpendicular of a triangle whose base is 5cm and the hypotenuse is 13cm. Also, find its area.
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC2 - AB2 = 2BC x ED
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: 9BP2 = 9BC2 + 4AC2
From given figure, In ∆ABC, If AC = 12 cm. then AB =?
Activity: From given figure, In ∆ABC, ∠ABC = 90°, ∠ACB = 30°
∴ ∠BAC = `square`
∴ ∆ABC is 30° – 60° – 90° triangle
∴ In ∆ABC by property of 30° – 60° – 90° triangle.
∴ AB = `1/2` AC and `square` = `sqrt(3)/2` AC
∴ `square` = `1/2 xx 12` and BC = `sqrt(3)/2 xx 12`
∴ `square` = 6 and BC = `6sqrt(3)`
Sides AB and BE of a right triangle, right-angled at B are of lengths 16 cm and 8 cm respectively. The length of the side of largest square FDGB that can be inscribed in the triangle ABE is ______.
In a quadrilateral ABCD, ∠A + ∠D = 90°. Prove that AC2 + BD2 = AD2 + BC2
[Hint: Produce AB and DC to meet at E.]
Lengths of sides of a triangle are 3 cm, 4 cm and 5 cm. The triangle is ______.
Two squares are congruent, if they have same ______.
In a triangle, sum of squares of two sides is equal to the square of the third side.
