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If the altitudes from two vertices of a triangle to the opposite sides are equal, prove that the triangle is isosceles.

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To prove
`triangleXYZ` is isosceles
Now, consider `triangleYZP` and `triangleYZQ`
Give we have `angle YQZ = ZPY = 90`
Common side: `YZ = ZY` And` ZQ = YP`
So, by RHS congruence criterion, we have `triangleYQZ cong triangleZPY`
Now, `angleQYZ = anglePYZ` [In congruent triangles corresponding parts are equal]
`angleXYZ = angleXZY`
`XZ = XY `[Opposite sides to equal angles are equal in a triangle]
Therefore `triangleXYZ` is isosceles
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If the altitude from one vertex of a triangle bisects the opposite side, then the triangle is isosceles. GIVEN : A A B C such that the altitude A D from A on the opposite side B C bisects B C i.e., B D=D Cdot TO PROVE : A B=A C i.e. the triangle A B C is isosceles.

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If the bisector of an angle of a triangle bisects the opposite side,prove that the triangle is isosceles.

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If the bisector of the vertical angle of a triangle bisects the base, prove that the triangle is isosceles.

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The length of the altitudes from the vertices A,B,C of a triangle ABC to opposite sides are 24,18,(72)/(5) respectively. If Delta is the area of the triangle ABC then root(3)(Delta) is equal to

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RD SHARMA-CONGRUENT TRIANGLE -All Questions
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  2. If A B C is an isosceles triangle with A B=A C . Prove that the ...

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  3. If the altitudes from two vertices of a triangle to the opposite sid...

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  4. A D and B C are equal perpendiculars to a line segment A B . Show that...

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  5. In A B C ,A B=A C , and the bisectors of angles B and C intersec...

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  6. In Figure, it is given that A B=E F ,B C=D E ,A B|B D and F E|C Edot P...

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  7. In Figure, it is given that A B=B C and A D=E C . Prove that A B E~=C ...

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  8. In Figure, l m and M is the mid-point of the line segment A Bdot Prov...

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  9. In Figure, line l is the bisector of angle A and B is any point on l ....

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  10. In Figure, O is the mid point of A B and C D . Prove that A O C~= B ...

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  11. Prove that measure of each angle of an equilateral triangle is 60^0d...

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  12. In Figure, it is given that A E=A D and B D=C Edot Prove that A E B~=...

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  13. In Figure, it is given that A B=C F ,E F=B D and /A F E=/C B Ddot Prov...

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  14. Two triangles are congruent if two sides and the included angle of o...

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  15. In DeltaABC ,/A=100^0 and A B=A Cdot Find /B and /C

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  16. Angles opposite to two equal sides of a triangle are equal. GIVEN : ...

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  17. In Figure, X and Y are two points on equal sides A B and A C of a A B ...

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  18. Prove that the angle between internal bisector of one base angle an...

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  19. In A B C and P Q R Figure, A B=P Q ,B C=Q R and C B and R Q are extend...

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  20. In Figure, the side B C of A B C is produced to form ray B D as shown....

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