Premium Online Home Tutors

3 Tutor System

Starting just at 265/hour

Prove that ar (ADX) = ar (ACY).

[Hint: Join CX.]

Answer :

It can be observed that, \(\triangle{ADX}\) and \(\triangle{ACX}\) lie on the same base AX and are between the same parallels AB and DC.

\(\therefore \) Area (\(\triangle{ADX}\)) = Area (\(\triangle{ACX}\)) ...(i)

Also, \(\triangle{ADY}\) and \(\triangle{ACX}\) lie on the same base AC and are between the same parallels AC and XY.

\(\therefore \) Area (\(\triangle{ACY}\)) = Arca (\(\triangle{ACX}\)) ...(ii)

From Equations (i) and (ii), we get,

Area (\(\triangle{ADX}\)) = Area (\(\triangle{ACY}\))

Hence, proved.

- In Figure, E is any point on median AD of a \(\triangle{ABC}\). Show that ar (ABE) = ar (ACE)
- In a triangle ABC, E is the mid-point of median AD. Show that ar (BED) = \(\frac{1}{4} \) ar (ABC).
- Show that the diagonals of a parallelogram divide it into four triangles of equal area.
- In Figure, ABC and ABD are two triangles on the same base AB. If line segment CD is bisected by AB at O, show that ar(ABC) = ar (ABD).
- D, E and F are respectively the mid-points of the sides BC, CA and AB of a \(\triangle{ABC}\). Show thati) BDEF is a parallelogram.ii) ar(DEF) = \(\frac{1}{4} \) ar(ABC) iii) ar(BDEF) = \(\frac{1}{2} \) ar(ABC)
- In Figure, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that:i) ar (DOC) = ar (AOB)ii) ar (DCB) = ar (ACB)iii) DA || CB or ABCD is a parallelogram.[Hint: From D and B, draw perpendiculars to AC.]
- D and E are points on sides AB and AC respectively of \(\triangle{ABC}\) such that ar (DBC) = ar (EBC). Prove that DE || BC.
- XY is a line parallel to side BC of a triangle ABC. If BE || AC and CF || AB meet XY at E and F respectively, show that ar (ABE) = ar (ACF)
- The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed (see Figure). Show that ar (ABCD) = ar (PBQR).[Hint : Join AC and PQ. Now compare ar (ACQ) and ar (APQ).]
- Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Prove that ar (AOD) = ar (BOC).
- In Figure, ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show thati) ar (ACB) = ar (ACF)ii) ar (AEDF) = ar (ABCDE).
- A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented.
- In Figure, AP || BQ || CR. Prove that ar (AQC) = ar (PBR).
- Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar (AOD) = ar (BOC). Prove that ABCD is a trapezium.
- In Figure, ar (DRC) = ar (DPC) and ar (BDP) = ar (ARC). Show that both the quadrilaterals ABCD and DCPR are trapeziums.

- NCERT solutions for class 9 maths chapter 1 Number Systems
- NCERT solutions for class 9 maths chapter 2 Polynomials
- NCERT solutions for class 9 maths chapter 3 Coordinate geometry
- NCERT solutions for class 9 maths chapter 4 Linear equations in two variables
- NCERT solutions for class 9 maths chapter 5 Introduction to Euclidean Geometry
- NCERT solutions for class 9 maths chapter 6 Lines and Angles
- NCERT solutions for class 9 maths chapter 7 Triangles
- NCERT solutions for class 9 maths chapter 8 Quadrilaterals
- NCERT solutions for class 9 maths chapter 9 Areas of parallelograms and triangles
- NCERT solutions for class 9 maths chapter 10 Circles
- NCERT solutions for class 9 maths chapter 11 Constructions
- NCERT solutions for class 9 maths chapter 12 Heron's Formula
- NCERT solutions for class 9 maths chapter 13 Surface areas and volumes
- NCERT solutions for class 9 maths chapter 14 Statistics
- NCERT solutions for class 9 maths chapter 15 Probability

- NCERT solutions for class 9 science chapter 1 Matter in our Surroundings
- NCERT solutions for class 9 science chapter 2 Is Matter Around Us Pure
- NCERT solutions for class 9 science chapter 3 Atoms and Molecules
- NCERT solutions for class 9 science chapter 4 Structure of the Atom
- NCERT solutions for class 9 science chapter 5 The Fundamental Unit of Life
- NCERT solutions for class 9 science chapter 6 Tissues and Fundamental unit of life
- NCERT solutions for class 9 science chapter 7 Diversity in Living Organisms
- NCERT solutions for class 9 science chapter 8 Motion
- NCERT solutions for class 9 science chapter 9 Force and Laws of Motion
- NCERT solutions for class 9 science chapter 10 Gravitation
- NCERT solutions for class 9 science chapter 11 Work and Energy
- NCERT solutions for class 9 science chapter 12 sound
- NCERT solutions for class 9 science chapter 13 Why do We Fall Ill
- NCERT solutions for class 9 science chapter 14 Natural Resources
- NCERT solutions for class 9 science chapter 15 Improvement in Food Resources