Advertisements
Advertisements
प्रश्न
In triangle ABC, ∠C=90°. Let BC= a, CA= b, AB= c and let 'p' be the length of the perpendicular from 'C' on AB, prove that:
1. cp = ab
2. `1/p^2=1/a^2+1/b^2`
Advertisements
उत्तर
1. Area of a triangle = (1/2) x Base x Height
A(ΔABC) = (1/2) x AB x CD
A(ΔABC) = (1/2) x cp .......(i)
Area of right angle triangle ABC = A(ΔABC) = (1/2) x AC x BC
A(ΔABC) = (1/2) x ba ........(ii)
From (i) and (ii)
cp=ba⇒ cp⇒ab ..........(iii)
2. We have,
cp=ab ..........From(iii)
p = ab/c
Square both sides of the equation.
We get, `p^2=(a^2b^2)/c^2`
`1/p^2=c^2/(a^2b^2)" ..................(iv).....[By invertendo]"`
In right angled triangle ABC,
AB2 = AC2 + BC2 ................[By Pythagoras’ theorem]
c2 = b2 + a2 ............(v)
`c^2/(a^2b^2) = b^2/(a^2b^2) + a^2/(a^2b^2)`..........[Dividing throughout by `a^2b^2`]
`c^2/(ab)^2 = 1/a^2 + 1/b^2` .........(iii)
`c^2/(cp)^2 = 1/a^2 + 1/b^2` ...........[From (ii) and (iii)]
`c^2/(c^2p^2) = 1/a^2 + 1/b^2`
`1/p^2 = 1/a^2 + 1/b^2`
APPEARS IN
संबंधित प्रश्न
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 a right triangle ABC right-angled at C, P and Q are the points on the sides CA and CB respectively, which divide these sides in the ratio 2 : 1. Prove that
`(i) 9 AQ^2 = 9 AC^2 + 4 BC^2`
`(ii) 9 BP^2 = 9 BC^2 + 4 AC^2`
`(iii) 9 (AQ^2 + BP^2 ) = 13 AB^2`
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. 7 cm, 24 cm, 25 cm
PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM . MR
Identify, with reason, if the following is a Pythagorean triplet.
(4, 9, 12)
Identify, with reason, if the following is a Pythagorean triplet.
(5, 12, 13)
In a trapezium ABCD, seg AB || seg DC seg BD ⊥ seg AD, seg AC ⊥ seg BC, If AD = 15, BC = 15 and AB = 25. Find A(▢ABCD)
A man goes 40 m due north and then 50 m due west. Find his distance from the starting point.
If the sides of the triangle are in the ratio 1: `sqrt2`: 1, show that is a right-angled triangle.
In a quadrilateral ABCD, ∠B = 90° and ∠D = 90°.
Prove that: 2AC2 - AB2 = BC2 + CD2 + DA2
In the following Figure ∠ACB= 90° and CD ⊥ AB, prove that CD2 = BD × AD
If the angles of a triangle are 30°, 60°, and 90°, then shown that the side opposite to 30° is half of the hypotenuse, and the side opposite to 60° is `sqrt(3)/2` times of the hypotenuse.
Find the value of (sin2 33 + sin2 57°)
In triangle PQR, angle Q = 90°, find: PR, if PQ = 8 cm and QR = 6 cm
Find the Pythagorean triplet from among the following set of numbers.
2, 4, 5
A man goes 10 m due east and then 24 m due north. Find the distance from the straight point.
In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.
SHow that ∠RTP = 90°.
Find the unknown side in the following triangles
If ΔABC ~ ΔPQR, `("ar" triangle "ABC")/("ar" triangle "PQR") = 9/4` and AB = 18 cm, then the length of PQ is ______.
The foot of a ladder is 6 m away from its wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its top reach?
