Advertisements
Advertisements
Question
ABC is a triangle, right-angled at B. M is a point on BC.
Prove that: AM2 + BC2 = AC2 + BM2
Solution
The pictorial form of the given problem is as follows:
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
First, we consider the ΔABM and applying Pythagoras theorem we get,
AM2 = AB2 + BM2
AB2 = AM2 - BM2 ...(i)
Now, we consider the ΔABC and applying Pythagoras theorem we get,
AC2 = AB2 + BC2
AB2 = AC2 - BC2 ...(ii)
From (i) and (ii) we get,
AM2 - BM2 = AC2 - BC2
AM2 + BC2 = AC2 + BM2
Hence, Proved.
APPEARS IN
RELATED QUESTIONS
In a right triangle ABC, right-angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle (in cm) is
(A) 4
(B) 3
(C) 2
(D) 1
Identify, with reason, if the following is a Pythagorean triplet.
(11, 60, 61)
In the given figure, M is the midpoint of QR. ∠PRQ = 90°. Prove that, PQ2 = 4PM2 – 3PR2.
Some question and their alternative answer are given. Select the correct alternative.
If a, b, and c are sides of a triangle and a2 + b2 = c2, name the type of triangle.
In ΔABC, Find the sides of the triangle, if:
- AB = ( x - 3 ) cm, BC = ( x + 4 ) cm and AC = ( x + 6 ) cm
- AB = x cm, BC = ( 4x + 4 ) cm and AC = ( 4x + 5) cm
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.
In figure AB = BC and AD is perpendicular to CD.
Prove that: AC2 = 2BC. DC.
In triangle ABC, ∠B = 90o and D is the mid-point of BC.
Prove that: AC2 = AD2 + 3CD2.
Choose the correct alternative:
In right-angled triangle PQR, if hypotenuse PR = 12 and PQ = 6, then what is the measure of ∠P?
Find the value of (sin2 33 + sin2 57°)
Triangle XYZ is right-angled at vertex Z. Calculate the length of YZ, if XY = 13 cm and XZ = 12 cm.
The sides of a certain triangle is given below. Find, which of them is right-triangle
16 cm, 20 cm, and 12 cm
In the given figure, angle ACP = ∠BDP = 90°, AC = 12 m, BD = 9 m and PA= PB = 15 m. Find:
(i) CP
(ii) PD
(iii) CD
Use the information given in the figure to find the length AD.
Find the Pythagorean triplet from among the following set of numbers.
9, 40, 41
In the figure, find AR
For going to a city B from city A, there is a route via city C such that AC ⊥ CB, AC = 2x km and CB = 2(x + 7) km. It is proposed to construct a 26 km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of the highway.
If the areas of two circles are the same, they are congruent.
If the hypotenuse of one right triangle is equal to the hypotenuse of another right triangle, then the triangles are congruent.
Jayanti takes shortest route to her home by walking diagonally across a rectangular park. The park measures 60 metres × 80 metres. How much shorter is the route across the park than the route around its edges?
