Advertisements
Advertisements
प्रश्न
In the given figure, ∠B = 90°, XY || BC, AB = 12 cm, AY = 8cm and AX : XB = 1 : 2 = AY : YC.
Find the lengths of AC and BC.
Advertisements
उत्तर
Given that AX : XB = 1 : 2 = AY : YC.
Let x be the common multiple for which this proportion gets satisfied.
So, AX = 1x and XB = 2x
AX + XB = 1x + 2x = 3x
⇒ AB = 3x .….(A - X - B)
⇒ 12 = 3x
⇒ x = 4
AX = 1x = 4 and XB = 2x = 2 × 4 = 8
Similarly,
AY = 1y and YC = 2y
AY = 8 …(given)
⇒ 8 = y
∴ YC = 2y = 2 × 8 = 16
∴ AC = AY + YC
AC = 8 + 16
AC = 24 cm
∆ABC is a right angled triangle. ...(Given)
∴ By Pythagoras Theorem, we get
⇒ AB2 + BC2 = AC2
⇒ BC2 = AC2 - AB2
⇒ BC2 = (24)2 - (12)2
⇒ BC2 = 576 - 144
⇒ BC2 = 432
⇒ BC = `sqrt(432)`
⇒ BC = `2 xx 2 xx 3sqrt3`
⇒ BC = `bb(12sqrt3 " cm")`
APPEARS IN
संबंधित प्रश्न
In the given figure, ∆ABC is an obtuse triangle, obtuse-angled at B. If AD ⊥ CB (produced) prove that AC2 = AB2 + BC2 + 2BC · BD.
PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM . MR
A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.
Identify, with reason, if the following is a Pythagorean triplet.
(3, 5, 4)
Diagonals of rhombus ABCD intersect each other at point O.
Prove that: OA2 + OC2 = 2AD2 - `"BD"^2/2`
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.
Triangle PQR is right-angled at vertex R. Calculate the length of PR, if: PQ = 34 cm and QR = 33.6 cm.
Find the Pythagorean triplet from among the following set of numbers.
2, 4, 5
Find the Pythagorean triplet from among the following set of numbers.
2, 6, 7
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?
40, 20, 30
From a point O in the interior of aΔABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove that: AF2 + BD2 + CE2 = OA2 + OB2 + OC2 - OD2 - OE2 - OF2
Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels at a speed of `(20 "km")/"hr"` and the second train travels at `(30 "km")/"hr"`. After 2 hours, what is the distance between them?
In a right angled triangle, the hypotenuse is the greatest side
If S is a point on side PQ of a ΔPQR such that PS = QS = RS, then ______.
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).
Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle.
The longest side of a right angled triangle is called its ______.
Points A and B are on the opposite edges of a pond as shown in the following figure. To find the distance between the two points, the surveyor makes a right-angled triangle as shown. Find the distance AB.
Two poles of 10 m and 15 m stand upright on a plane ground. If the distance between the tops is 13 m, find the distance between their feet.
