Home
Class 10
MATHS
P and Q are the mid-points of the sides ...

P and Q are the mid-points of the sides CA and CB respectively of a `triangleABC` , right angled at C, prove that.
(i) `4AQ^(2)=4AC^(2)+BC^(2)` (ii) `4BP^(2)=4BC^(2)+AC^(2)` (iii) `4(AQ^(2)+BP^(2))5AB^(2)`

Text Solution

Verified by Experts

(i) `triangleAQC , angleC= 90^(@)`
Using Pythagoras theorem
`AQ^(2)=AC^(2)+QC^(2)`
`AQ^(2)= AC^(2) + ((BC)/2)^(2)`
(Q is the mid-point of BC)
`AQ^(2)= AC^(2)+(BC^(2))/4`
`4AQ^(2)= 4AC^(2)+BC^(2)` …(1) Hence proved.
(ii) `triangleBPC, angleC=90^(@)` ,
Using Pythagoras theorem, we have
`BP^(2)=BC^(2) + PC^(2)` ,
`BP^(2)=BC^(2)+ ((AC)/2)^(2)` (since P is the mid-point of AC)
`Rightarrow " " 4 BP^(2)= 4BC^(2)+AC^(2)` ...(2) Hence proved.
(iii) Adding (1) and (2) we get
`4AQ^(2)+4BP^(2)=4AC^(2)+BC^(2)+4BC^(2)+AC^(2)`
`4(AQ^(2)+BP^(2))= 5(AC^(2)+BC^(2))`
`Rightarrow " " 4(AQ^(2)+BP^(2))= 5AB^(2) " " ( AC^(2) +BC^(2)=AB^(2))` Hence proved.
Promotional Banner

Topper's Solved these Questions

  • TRIANGLES

    NAGEEN PRAKASHAN ENGLISH|Exercise Problems From NCERT/ Exemplar|14 Videos

  • TRIANGLES

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercise 6a|24 Videos

  • STATISTICS

    NAGEEN PRAKASHAN ENGLISH|Exercise Revision Exercise Long Answer Questions|4 Videos

  • VOLUME AND SURFACE AREA OF SOLIDS

    NAGEEN PRAKASHAN ENGLISH|Exercise Revisions Exercise Long Answer Questions|5 Videos

Similar Questions

Explore conceptually related problems

P and Q are the mid-points of the CA and CD respectively of a triangle ABC, right angled at C. Prove that: 4AQ^2 = 4AC^2 + BC^2 , 4BP^2 = 4BC^2 + AC^2 , and 4(AQ^2 + BP^2) = 5AB^2 .

M and N are the mid-points of the sides QR and PQ respectively of a DeltaPQR , right-angled at Q. Prove that : (i) PM^(2)+RN^(2)=5MN^(2) (ii) 4PM^(2)=4PQ^(2)+QR^(2) (iii) 4RN^(2)=PQ^(2)+4QR^(2) (iv) 4(PM^(2)+RN^(2))=5PR^(2)

In triangleABC , angleA= 60^(@) prove that BC^(2)= AB^(2)+AC^(2)- AB . AC

P and Q are points on the sides AB and AC respectively of a triangleABC . If AP= 2cm, PB = 4cm AQ= 3cm and QC= 6cm. Show that BC= 3PQ.

If P and Q are points on the sides AB and AC respeactively of triangleABC , if PQ||BC,l AP= 2 cm , AB = 6 cm and AC= 9 cm find AQ.

ABC is a triangle, right angled at B, M is a point on BC. Prove that : AM^(2) + BC^(2) = AC^(2) + BM^(2)

In the following figure, OP, OQ and OR are drawn perpendiculars to the sides BC, CA and AB repectively of triangle ABC. Prove that : AR^(2)+BP^(2)+CQ^(2)=AQ^(2)+CP^(2)+BR^(2)

In a rhombus ABCD, prove that AC^(2) + BD^(2) = 4AB^(2)

In an acute angled triangle ABC, AD is the median in it. Prove that : AD^(2) = (AB^(2))/2 + (AC^(2))/2 - (BC^(2))/4

In a right triangle ABC, right angled at A, AD is drawn perpendicular to BC. Prove that: AB^(2)-BD^(2)= AC^(2)-CD^(2)