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
P and Q are the mid-points of the sides CA and CB respectively of a ∆ABC, right angled at C. Prove that:
`(i) 4AQ^2 = 4AC^2 + BC^2`
`(ii) 4BP^2 = 4BC^2 + AC^2`
`(iii) (4AQ^2 + BP^2 ) = 5AB^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. 13 cm, 12 cm, 5 cm
ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2, prove that ABC is a right triangle.
In the given figure, M is the midpoint of QR. ∠PRQ = 90°. Prove that, PQ2 = 4PM2 – 3PR2.
In right angle ΔABC, if ∠B = 90°, AB = 6, BC = 8, then find AC.
If the sides of the triangle are in the ratio 1: `sqrt2`: 1, show that is a right-angled triangle.
In triangle ABC, AB = AC and BD is perpendicular to AC.
Prove that: BD2 − CD2 = 2CD × AD
The sides of a certain triangle is given below. Find, which of them is right-triangle
6 m, 9 m, and 13 m
In the given figure, angle ACB = 90° = angle ACD. If AB = 10 m, BC = 6 cm and AD = 17 cm, find :
(i) AC
(ii) CD
Find the Pythagorean triplet from among the following set of numbers.
3, 4, 5
The sides of the triangle are given below. Find out which one is the right-angled triangle?
40, 20, 30
From the given figure, find the length of hypotenuse AC and the perimeter of ∆ABC.
A man goes 10 m due east and then 24 m due north. Find the distance from the straight point.
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC2 = AD2 + BC x DE + `(1)/(4)"BC"^2`
AD is perpendicular to the side BC of an equilateral ΔABC. Prove that 4AD2 = 3AB2.
In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.
SHow that ∠RTP = 90°.
In the given figure. PQ = PS, P =R = 90°. RS = 20 cm and QR = 21 cm. Find the length of PQ correct to two decimal places.
In a right-angled triangle ABC, if angle B = 90°, BC = 3 cm and AC = 5 cm, then the length of side AB is ______.
The hypotenuse (in cm) of a right angled triangle is 6 cm more than twice the length of the shortest side. If the length of third side is 6 cm less than thrice the length of shortest side, then find the dimensions of the triangle.
